1 cs 4700: foundations of artificial intelligence bart selman [email protected] module:...
TRANSCRIPT
1
CS 4700Foundations of Artificial Intelligence
Bart Selmanselmancscornelledu
Module Knowledge Reasoning and PlanningPart 1
Logical AgentsRampN Chapter 7
2
A Model-Based Agent
Requires Knowledge and Reasoning
3
Knowledge and Reasoning
Knowledge and Reasoning humans are very good at acquiring new information by combining raw knowledge experience with reasoning AI-slogan ldquoKnowledge is powerrdquo (or ldquoData is powerrdquo)
Examples Medical diagnosis --- physician diagnosing a patient infers what disease based on the knowledge heshe acquired as a student textbooks prior cases Common sense knowledge reasoning --- common everyday assumptions inferences eg ldquolecture starts at fourrdquo infer pm not am when traveling I assume there is some way to get from the airport to the hotel
4
Logical agents Agents with some representation of the complex knowledge about the world its environment and uses inference to derive new information from that knowledge combined with new inputs (eg via perception)
Key issues1- Representation of knowledge
What form Meaning semantics2- Reasoning and inference
processes Efficiency
5
Knowledge-base Agents
Key issuesndash Representation of knowledge knowledge basendash Reasoning processes inferencereasoning
() called Knowledge Representation (KR) language
Knowledge base = set of sentences in a formal language representing facts about the world()
6
Knowledge basesKey aspects
ndash How to add sentences to the knowledge base ndash How to query the knowledge base
Both tasks may involve inference ndash ie how to derive new sentences from old sentences
Logical agents ndash inference must obey the fundamental requirement that when one asks a question to the knowledge base the answer should follow from what has been told to the knowledge base previously (In other words the inference process should not ldquomake thingsrdquo uphellip)
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
2
A Model-Based Agent
Requires Knowledge and Reasoning
3
Knowledge and Reasoning
Knowledge and Reasoning humans are very good at acquiring new information by combining raw knowledge experience with reasoning AI-slogan ldquoKnowledge is powerrdquo (or ldquoData is powerrdquo)
Examples Medical diagnosis --- physician diagnosing a patient infers what disease based on the knowledge heshe acquired as a student textbooks prior cases Common sense knowledge reasoning --- common everyday assumptions inferences eg ldquolecture starts at fourrdquo infer pm not am when traveling I assume there is some way to get from the airport to the hotel
4
Logical agents Agents with some representation of the complex knowledge about the world its environment and uses inference to derive new information from that knowledge combined with new inputs (eg via perception)
Key issues1- Representation of knowledge
What form Meaning semantics2- Reasoning and inference
processes Efficiency
5
Knowledge-base Agents
Key issuesndash Representation of knowledge knowledge basendash Reasoning processes inferencereasoning
() called Knowledge Representation (KR) language
Knowledge base = set of sentences in a formal language representing facts about the world()
6
Knowledge basesKey aspects
ndash How to add sentences to the knowledge base ndash How to query the knowledge base
Both tasks may involve inference ndash ie how to derive new sentences from old sentences
Logical agents ndash inference must obey the fundamental requirement that when one asks a question to the knowledge base the answer should follow from what has been told to the knowledge base previously (In other words the inference process should not ldquomake thingsrdquo uphellip)
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
3
Knowledge and Reasoning
Knowledge and Reasoning humans are very good at acquiring new information by combining raw knowledge experience with reasoning AI-slogan ldquoKnowledge is powerrdquo (or ldquoData is powerrdquo)
Examples Medical diagnosis --- physician diagnosing a patient infers what disease based on the knowledge heshe acquired as a student textbooks prior cases Common sense knowledge reasoning --- common everyday assumptions inferences eg ldquolecture starts at fourrdquo infer pm not am when traveling I assume there is some way to get from the airport to the hotel
4
Logical agents Agents with some representation of the complex knowledge about the world its environment and uses inference to derive new information from that knowledge combined with new inputs (eg via perception)
Key issues1- Representation of knowledge
What form Meaning semantics2- Reasoning and inference
processes Efficiency
5
Knowledge-base Agents
Key issuesndash Representation of knowledge knowledge basendash Reasoning processes inferencereasoning
() called Knowledge Representation (KR) language
Knowledge base = set of sentences in a formal language representing facts about the world()
6
Knowledge basesKey aspects
ndash How to add sentences to the knowledge base ndash How to query the knowledge base
Both tasks may involve inference ndash ie how to derive new sentences from old sentences
Logical agents ndash inference must obey the fundamental requirement that when one asks a question to the knowledge base the answer should follow from what has been told to the knowledge base previously (In other words the inference process should not ldquomake thingsrdquo uphellip)
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
4
Logical agents Agents with some representation of the complex knowledge about the world its environment and uses inference to derive new information from that knowledge combined with new inputs (eg via perception)
Key issues1- Representation of knowledge
What form Meaning semantics2- Reasoning and inference
processes Efficiency
5
Knowledge-base Agents
Key issuesndash Representation of knowledge knowledge basendash Reasoning processes inferencereasoning
() called Knowledge Representation (KR) language
Knowledge base = set of sentences in a formal language representing facts about the world()
6
Knowledge basesKey aspects
ndash How to add sentences to the knowledge base ndash How to query the knowledge base
Both tasks may involve inference ndash ie how to derive new sentences from old sentences
Logical agents ndash inference must obey the fundamental requirement that when one asks a question to the knowledge base the answer should follow from what has been told to the knowledge base previously (In other words the inference process should not ldquomake thingsrdquo uphellip)
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
5
Knowledge-base Agents
Key issuesndash Representation of knowledge knowledge basendash Reasoning processes inferencereasoning
() called Knowledge Representation (KR) language
Knowledge base = set of sentences in a formal language representing facts about the world()
6
Knowledge basesKey aspects
ndash How to add sentences to the knowledge base ndash How to query the knowledge base
Both tasks may involve inference ndash ie how to derive new sentences from old sentences
Logical agents ndash inference must obey the fundamental requirement that when one asks a question to the knowledge base the answer should follow from what has been told to the knowledge base previously (In other words the inference process should not ldquomake thingsrdquo uphellip)
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
6
Knowledge basesKey aspects
ndash How to add sentences to the knowledge base ndash How to query the knowledge base
Both tasks may involve inference ndash ie how to derive new sentences from old sentences
Logical agents ndash inference must obey the fundamental requirement that when one asks a question to the knowledge base the answer should follow from what has been told to the knowledge base previously (In other words the inference process should not ldquomake thingsrdquo uphellip)
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
A simple knowledge-based agent
The agent must be able tondash Represent states actions etcndash Incorporate new perceptsndash Update internal representations of the worldndash Deduce hidden properties of the worldndash Deduce appropriate actionsndash
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
8
KR language candidate logical language (propositional first-order) combined with a logical inference mechanism
How close to human thought (mental-models Johnson-Laird)
What is ldquothe language of thoughtrdquo
Why not use natural language (eg English)
We want clear syntax amp semantics (well-definedmeaning) and mechanism to infer new informationSoln Use a formal language
Greeks Boole Frege --- Rational thought Logic
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
9
Consider to-the-right-of(xy)
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
10
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
12
Procedural style
Knowledge-based alternative
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
13
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
14
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
15The ldquosymbol grounding problemrdquo
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
16
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
17
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
18
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
19Ie Models(KB) Models( )
Note KB defines exactly the set of worlds we are interested in
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
20
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
21
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
22
Note (1) This was Aristotlersquos original goal ---Construct infallible arguments based purelyon the form of statements --- not on the ldquomeaningrdquoof individual propositions(2) Sets of models can be exponential size or worsecompared to symbolic inference (deduction)
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
23
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
24
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
25
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
26
Intermezzo Reflections on Inference Knowledge and Data ca 2012
In the logical agent perspective we take the ldquoknowledge-basedrdquo approach
Wersquoll have a knowledge base capturing our world knowledge in a formal language Wersquoll use an inference procedure to derive new information
Representation has well-defined syntax and semantics meaning
How far are actual AI systems in the knowledge reasoning paradigm
Specifically where do Deep Blue Watson and Siri fit
What about IR Google Translate and Googlersquos Knowledge Graph
And ldquoshallow semanticsrdquo Semantic Web
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
27
WatsonIBMrsquos Jeopardy playing system
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
28
Chris Welty from IBM on Watson and ldquounderstandingrdquo
httpspectrumieeeorgpodcastat-workinnovationwhat-is-toronto
See 4 min mark for discussion on ldquounderstandingrdquoTill around min 11
29
WatsonKey aspects
1) Limited number of categories of types of questions
2) Well-defined knowledge sources (Wikipedia Encyclopedias
Dictionaries etc) Not WWW (Contrast Google Knowl Graph)
3) Around 70 ldquoexpertrdquo modules that handle different aspects of the possible question and answers 1000+ hypotheses
4) Final result based on a confidence vote
5) Limited language parsing Lot based on individual words (like IR)
6) Form of IR ldquo+rdquo (parsing limited knowledge graph ldquobooks have authorsrdquo ldquomovie has actors and directorrdquo ldquoverb represents an actionrdquo etc )
7) Some game theory to reason about whether to answer
To some extent Watson ldquounderstandsrdquo(Itrsquos actually a bit of a problem for IBMhellip Clients expect too much)
31
Semantic Web
Datagov in process of converting CSV data into Resource Description Framework
ndash RPI Tetherless World Constellation - Jim Hendler Berners-Lee
ndash RDF - triples (facts) class subclass
ndash OWL - defined classes constraints
Directly grows out of KR work in AAAI community 1985 - 2005
ndash Ron Brachman Hector Levesque Deborah McGuinness Alex Borgida Peter Patel-Schneider Franz Baader Francesco Donini
32
AI Knowledge-Data-
Inference Triangle
2012
KnowledgeIntensive
DataIntensive
InferenceSearch
Intensive
Common Sense
Watson
Google Search (IR)
Googlersquos Knowl Graph
SiriVerification
NLU
Computer Vision
Deep Blue
Robbinrsquos ConjM
achine LearningSemantic Web
20+yr GAP
Google Transl
4-color thm
33
Back to Knowledge-Based Agentshellip
34
Illustrative example Wumpus World
Performance measurendash gold +1000 ndash death -1000 (falling into a pit or being eaten by the wumpus)ndash -1 per step -10 for using the arrow
Environmentndash Rooms squares connected by doors ndash Squares adjacent to wumpus are smellyndash Squares adjacent to pit are breezyndash Glitter iff gold is in the same squarendash Shooting kills wumpus if you are facing itndash Shooting uses up the only arrowndash Grabbing picks up gold if in same squarendash Releasing drops the gold in same squarendash Randomly generated at start of game Wumpus only senses current room
Sensors Stench Breeze Glitter Bump Scream [perceptual inputs]Actuators Left turn Right turn Forward Grab Release Shoot
(Somewhat whimsical)
35
Wumpus world characterization
Fully Observable No ndash only local perception
Deterministic Yes ndash outcomes exactly specified
Static Yes ndash Wumpus and Pits do not move
Discrete Yes
Single-agent Yes ndash Wumpus is essentially a ldquonatural featurerdquo
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
31
Semantic Web
Datagov in process of converting CSV data into Resource Description Framework
ndash RPI Tetherless World Constellation - Jim Hendler Berners-Lee
ndash RDF - triples (facts) class subclass
ndash OWL - defined classes constraints
Directly grows out of KR work in AAAI community 1985 - 2005
ndash Ron Brachman Hector Levesque Deborah McGuinness Alex Borgida Peter Patel-Schneider Franz Baader Francesco Donini
32
AI Knowledge-Data-
Inference Triangle
2012
KnowledgeIntensive
DataIntensive
InferenceSearch
Intensive
Common Sense
Watson
Google Search (IR)
Googlersquos Knowl Graph
SiriVerification
NLU
Computer Vision
Deep Blue
Robbinrsquos ConjM
achine LearningSemantic Web
20+yr GAP
Google Transl
4-color thm
33
Back to Knowledge-Based Agentshellip
34
Illustrative example Wumpus World
Performance measurendash gold +1000 ndash death -1000 (falling into a pit or being eaten by the wumpus)ndash -1 per step -10 for using the arrow
Environmentndash Rooms squares connected by doors ndash Squares adjacent to wumpus are smellyndash Squares adjacent to pit are breezyndash Glitter iff gold is in the same squarendash Shooting kills wumpus if you are facing itndash Shooting uses up the only arrowndash Grabbing picks up gold if in same squarendash Releasing drops the gold in same squarendash Randomly generated at start of game Wumpus only senses current room
Sensors Stench Breeze Glitter Bump Scream [perceptual inputs]Actuators Left turn Right turn Forward Grab Release Shoot
(Somewhat whimsical)
35
Wumpus world characterization
Fully Observable No ndash only local perception
Deterministic Yes ndash outcomes exactly specified
Static Yes ndash Wumpus and Pits do not move
Discrete Yes
Single-agent Yes ndash Wumpus is essentially a ldquonatural featurerdquo
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
32
AI Knowledge-Data-
Inference Triangle
2012
KnowledgeIntensive
DataIntensive
InferenceSearch
Intensive
Common Sense
Watson
Google Search (IR)
Googlersquos Knowl Graph
SiriVerification
NLU
Computer Vision
Deep Blue
Robbinrsquos ConjM
achine LearningSemantic Web
20+yr GAP
Google Transl
4-color thm
33
Back to Knowledge-Based Agentshellip
34
Illustrative example Wumpus World
Performance measurendash gold +1000 ndash death -1000 (falling into a pit or being eaten by the wumpus)ndash -1 per step -10 for using the arrow
Environmentndash Rooms squares connected by doors ndash Squares adjacent to wumpus are smellyndash Squares adjacent to pit are breezyndash Glitter iff gold is in the same squarendash Shooting kills wumpus if you are facing itndash Shooting uses up the only arrowndash Grabbing picks up gold if in same squarendash Releasing drops the gold in same squarendash Randomly generated at start of game Wumpus only senses current room
Sensors Stench Breeze Glitter Bump Scream [perceptual inputs]Actuators Left turn Right turn Forward Grab Release Shoot
(Somewhat whimsical)
35
Wumpus world characterization
Fully Observable No ndash only local perception
Deterministic Yes ndash outcomes exactly specified
Static Yes ndash Wumpus and Pits do not move
Discrete Yes
Single-agent Yes ndash Wumpus is essentially a ldquonatural featurerdquo
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
33
Back to Knowledge-Based Agentshellip
34
Illustrative example Wumpus World
Performance measurendash gold +1000 ndash death -1000 (falling into a pit or being eaten by the wumpus)ndash -1 per step -10 for using the arrow
Environmentndash Rooms squares connected by doors ndash Squares adjacent to wumpus are smellyndash Squares adjacent to pit are breezyndash Glitter iff gold is in the same squarendash Shooting kills wumpus if you are facing itndash Shooting uses up the only arrowndash Grabbing picks up gold if in same squarendash Releasing drops the gold in same squarendash Randomly generated at start of game Wumpus only senses current room
Sensors Stench Breeze Glitter Bump Scream [perceptual inputs]Actuators Left turn Right turn Forward Grab Release Shoot
(Somewhat whimsical)
35
Wumpus world characterization
Fully Observable No ndash only local perception
Deterministic Yes ndash outcomes exactly specified
Static Yes ndash Wumpus and Pits do not move
Discrete Yes
Single-agent Yes ndash Wumpus is essentially a ldquonatural featurerdquo
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
34
Illustrative example Wumpus World
Performance measurendash gold +1000 ndash death -1000 (falling into a pit or being eaten by the wumpus)ndash -1 per step -10 for using the arrow
Environmentndash Rooms squares connected by doors ndash Squares adjacent to wumpus are smellyndash Squares adjacent to pit are breezyndash Glitter iff gold is in the same squarendash Shooting kills wumpus if you are facing itndash Shooting uses up the only arrowndash Grabbing picks up gold if in same squarendash Releasing drops the gold in same squarendash Randomly generated at start of game Wumpus only senses current room
Sensors Stench Breeze Glitter Bump Scream [perceptual inputs]Actuators Left turn Right turn Forward Grab Release Shoot
(Somewhat whimsical)
35
Wumpus world characterization
Fully Observable No ndash only local perception
Deterministic Yes ndash outcomes exactly specified
Static Yes ndash Wumpus and Pits do not move
Discrete Yes
Single-agent Yes ndash Wumpus is essentially a ldquonatural featurerdquo
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
35
Wumpus world characterization
Fully Observable No ndash only local perception
Deterministic Yes ndash outcomes exactly specified
Static Yes ndash Wumpus and Pits do not move
Discrete Yes
Single-agent Yes ndash Wumpus is essentially a ldquonatural featurerdquo
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
36
Exploring a wumpus world
Stench Breeze Glitter Bump Scream
None none none none none
The knowledge base of the agent consists of the rules of the Wumpus world plus the percept ldquonothingrdquo in [11]
Boolean perceptfeature valueslt0 0 0 0 0gt
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
37
Stench Breeze Glitter Bump Scream
None none none none none
T=0 The KB of the agent consists of the rules of the Wumpus world plusthe percept ldquonothingrdquo in [11]By inference the agentrsquos knowledge base also has the information that[21] and [12] are okayAdded as propositions
World ldquoknownrdquo to agentat time = 0
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
38
Further exploration
Stench Breeze Glitter Bump Scream
T = 1 What followsPit(22) or Pit(31)
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
Where next
T = 0T = 1
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
39
S
Where is Wumpus
Wumpus cannot be in (11) or in (22) (Why) Wumpus in (13)Not breeze in (12) no pit in (22) but we know there is pit in (22) or (31) pit in (31)
P
P
1 2 3 4
1
2
3
4
Stench none none none none
P
W
PS
T=3
Stench Breeze Glitter Bump Scream
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
40
P
W
P
We reasoned about the possible states the Wumpus world can be in given our percepts and our knowledge of the rules of the Wumpus world Ie the content of KB at T=3
Essence of logical reasoning Given all we know Pit_in_(31) holds
(ldquoThe world cannot be differentrdquo)
What follows is what holds true in all those worlds that satisfy what is known at that time T=3 about the particular Wumpus world we are in
Models(KB) Models(P_in_(31))
Example property P_in_(31)
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
41
Formally Entailment
Situation after detecting nothing in [11] moving right breeze in [21] Ie T=1
Consider possible models for KB with respect to the cells (12) (22) and (31) with respect to the existence or non existence of pits
3 Boolean choices 8 possible interpretations
(enumerate all the models or
ldquopossible worldsrdquo wrt Pitt location)
Knowledge Base (KB) in the Wumpus World Rules of the wumpus world + new percepts
T = 1
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
42
KB = Wumpus-world rules + observations (T=1)
Is KB consistent with all8 possible worlds Worlds
that violate KB(are inconsistentwith what weknow)
Q Why does world violate KB
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
43
Entailment in Wumpus World
KB = Wumpus-world rules + observations
α1 = [12] has no pit KB α1
ndash In every model in which KB is true α1 is True (proved by ldquomodel checkingrdquo)
Models of the KB and α1
So KB definesall worlds thatwe hold possibleQueries we want to know the properties of those worldsThatrsquos how the semantics of logical entailment is defined
Note alpha_1holds in more models than KBThatrsquos OK but we donrsquot care aboutthose worlds
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
44
Wumpus modelsKB = wumpus-world rules + observations
α2 = [22] has no pit this is only True in some
of the models for which KB is True therefore KB α2
Model Checking
Models of α2
A model of KB where does NOT hold α2
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
45
Entailment vialdquoModel Checkingrdquo
Inference by Model checking ndash We enumerate all the KB models and check if α1 and α2 are
True in all the models (which implies that we can only use it
when we have a finite number of models)
Ie using semantics directly
Models(KB) Models( )
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
46
Example redux More formal
Stench Breeze Glitter Bump Scream
None none none none none
V
A ndash agentV ndash visitedB - breeze
AB P
P
None breeze none none none
How do we actually encode backgroundknowledge and percepts in formal language
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
47
Wumpus World KBDefine propositions
Let Pij be true if there is a pit in [i j]
Let Bij be true if there is a breeze in [i j]
Sentence 1 (R1) P11
Sentence 2 (R2) B11
Sentence 3 (R3) B21
Pits cause breezes in adjacent squaresrdquo
Sentence 4 (R4) B11 (P12 P21)
Sentence 5 (R5) B21 (P11 P22 P31)etc
Notes (1) one such statement about Breeze for each square (2) similar statements about Wumpus and stench
and Gold and glitter (Need more propositional letters)
[Given][Observation T = 0][Observation T = 1]
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
48
What about Time What about ActionsIs Time represented NoCan include time in propositions
Explicit time Pijt Bijt Lijt etc
Many more props O(TN2) (Lijt for agent at (ij) at time t)
Now we can also model actions use props Move(i j k l t) Eg Move(1 1 2 1 0)What knowledge axiom(s) capture(s) the effect of an Agent
move
Move(i j k l t) ( L(i j t+1) AElig L(k l t+1))
Is this it What about i j k and lWhat about Agent location at time t
)
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
49
Improved
Move(i j k l t) (L(i j t) AElig L(i j t+1) AElig L(k l t+1)) For all tuples (i j k l) that represent legitimate possible moves Eg (1 1 2 1) or (1 1 1 2)
Still some remaining subtleties when representing time andactions What happens to propositions at time t+1 compared to attime t that are not involved in any actionEg P(1 3 3) is derived at some point What about P(1 3 4) True or False RampN suggests having P as an ldquoatemporal varrdquo since it cannot change over
time Nevertheless we have many other vars that can change over time called ldquofluentsrdquo
Values of propositions not involved in any action should not change ldquoThe Frame Problemrdquo Frame Axioms RampN 771
Move implies a change in the world statea change in the world state implies a move occurred
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
50
Successor-State Axioms
Ft+1 (ActionCausesFt Ccedil (F(t) ActionCausesNotFt))
Axiom schemaF is a fluent (prop that can change over time)
For example
ie L_11 was ldquoas beforerdquo with [no movement action or bump into wall] or resulted from some action (movement into L_11
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
51
Some example inferencesSection 771 RampN
Actions and inputs up to time 6Note includes turns
Define ldquoOKrdquo
In milliseconds with modern SAT solver
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
52
Alternative formulation Situation CalculusRampN 1042
No explicit time Actions are what changes the worldfrom ldquosituationrdquo to ldquosituationrdquo More elegant but still need frame axioms to capture what stays the sameInherent with many representation formalisms ldquophysicalrdquopersistance does not come for free (and probably shouldnrsquot)
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
53
Inference by enumeration ldquomodel checkingrdquoStyle I
The goal of logical inference is to decide whether KB α for some For example given the rules of the Wumpus World is P22 entailed Relevant propositional symbols
R1 P11
R2 B11
R3 B21
Pits cause breezes in adjacent squaresR4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
Inference by enumeration We have 7 relevant symbols Therefore 27 = 128 interpretations Need to check if P22 is true in all of the KB models(interpretations that satisfy KB sentences)
Q KB has many more symbols Why can we restrict ourselves to these symbols here But be careful typically we canrsquot
Models(KB) Models( P22 )
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
54
All equivalentProp FO Logic1) KB sup2 reg
2) M(KB) micro M(reg)
3) sup2 (KB ) reg) deduction thm RampN 75
4) KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo enumeration (style I RampN 744)
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
entailment
5) (KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
55
Proof techniques
M(KB) micro M(reg)
KB ` reg
by defn semantic proofs truth tablesldquomodel checkingrdquo (style I RampN 744) Done
soundness and completenesslogical deduction symbol pushingproof by inference rules (style II)eg modus ponens (RampN 751)
(KB AElig reg) is inconsistent Proof by contradictionuse CNF clausal formResolution (style III RampN 75)SAT solvers (style IV RampN 76) most effective
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
56
Aside
Standard syntax and semantics for propositionallogic (CS-2800 see 741 and 742)
Syntax
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
57
Semantics
Note Truth value of a sentence is built from its parts ldquocompositional semanticsrdquo
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
58
Logical equivalences
()
() key to go to clausal (Conjunctive Normal Form)Implication for ldquohumansrdquo clauses for machinesde Morgan laws also very useful in going to clausal form
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
59
Style II Proof by inference rulesModus Ponens (MP)
P
P
Wumpus worldat T = 1
V AB
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
How can we show that KR sup2 P12 R4 B11 (P12 P21)It follows (bicond elim)
B11 ) (P12 P21) AElig (P12 P21) ) B11
And-elimination (751 RampN)
(P12 P21) ) B11
By contrapositive
B11 ) ( (P12 Ccedil P21))Thus (de Morgan)
B11 ) (P12 AElig P21)By Modus Ponens using R2 we get
(P12 AElig P21)Finally by And-elimination we get
P12
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
60
Length of Proofs
Why bother with inference rules We could always use a truth table
to check the validity of a conclusion from a set of premises
But resulting proof can be much shorter than truth table method
Consider KBp_1 p_1 p_2 p_2 p_3 hellip p_(n-1) p_n
To prove conclusion p_n
Inference rules Truth table n-1 MP steps 2n
Key open question Is there always a short proof for any validconclusion Probably not The NP vs co-NP question(The closely related P vs NP question carries a $1M prize)
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
61
Style III Resolution
P
P
Wumpus worldat T = 1
V AB
First we need a conversion to ConjunctiveNormal Form (CNF) or Clausal Form
Letrsquos consider converting R4 in clausal form
R4 B11 (P12 P21)We have
B11 ) (P12 Ccedil P21)which gives (implication elimination)
( B11 Ccedil P12
Ccedil P21)
Also
(P12 P21) ) B11
which gives
( (P12 Ccedil P21) Ccedil B11) Thus
( P12 AElig P21) Ccedil B11 leaving
( P12 Ccedil B11 )
( P21 Ccedil B11 )
(note clauses in red)
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
62
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
KB at T = 1R1 P11
R2 B11
R3 B21
R4 B11 (P12 P21)
R5 B21 (P11 P22 P31)
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
63
How can we show that KR sup2 P12
Proof by contradiction Need to show that (KB AElig P12) is inconsistent (unsatisfiable)
Resolution rule
(reg Ccedil p) and (macr Ccedil p)
gives resolvent (logically valid conclusion)
(reg Ccedil macr)
If we can reach the empty clause then KBis inconsistent (And vice versa)
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
64
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21
Show that (KB AElig P12) is inconsistent (unsatisfiable)
R4b with P12 resolves to B11which with R2 resolves to the empty clause So we can conclude KB sup2 P12 (make sure you use ldquowhat you want to proverdquo)
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
65
P
P
Wumpus worldat T = 1
V AB
KB at T=1 in clausal form
R1 P11
R2 B11
R3 B21
R4a B11 Ccedil P12
Ccedil P21
R4b P12 Ccedil B11
R4c P21 Ccedil B11
R5a B21 Ccedil P11 Ccedil P22 Ccedil P31
R5b P11 Ccedil B21
R5c P22 Ccedil B21
R5d P31 Ccedil B21 Note that R5a resolved with R1 andthen resolved with R3 gives (P22 Ccedil P31)
Almost therehellip to show KB sup2 (P22 Ccedil P31) we need to show KB AElig ( (P22 Ccedil P31)) is inconsistent So show KB AElig P22 AElig P31 is inconsistentThis follows from (P22 Ccedil P31) because in two more resolution steps we get the empty clause (a contradiction)
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
66
Length of ProofsWhat is hard for resolution
Consider Given a fixed pos int N
(P(i1) Ccedil P(i2) Ccedil hellip P(iN)) for i = 1 hellip N+1
( P(ij) Ccedil P(irsquoj)) for j= 1 hellip N i = 1 hellip N+1 irsquo = 1 hellipN+1 j == jrsquo
Think of P(ij) for ldquoobject i in location jrdquo
What does this encode
Pigeon hole problemhellip
Provable requires exponential number of resolutionsteps to reach empty clause (Haken 1985) Method ldquocanrsquot countrdquo
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
67
Style IV SAT SolversInstead of using resolution to show that
KB AElig reg
modern Satisfiability (SAT) solvers operating on the clausal formare much more efficient
The SAT solvers treat the set of clauses as a set of constraints (disjunctions) on Boolean variables ie a CSP problem Current solvers are very powerful Can handle 1 Million+ variables and several millions of clauses
Systematic Davis Putnam (DPLL) + series of improvementsStochastic local search WalkSAT (issue)
See RampN 76 ldquoIronicallyrdquo we are back to semantic model checking but way more clever than basic truth assignment enumeration (exponentially faster)
is inconsistent
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
68
DPLL improvements
1) Component analysis (disjoint sets of constraints Problem decomposition)
2) Clever variable and value ordering (eg degree heuristics)
3) Intelligent backtracking and clause learning (conflict learning)
4) Random restarts (heavy tails in search spaceshellip)
5) Clever data structures
Backtracking + hellip
1+ Million Boolean vars amp 10+ Million clauseconstraintsare feasible nowadays (eg Minisat solver)
Has changed the world of verification (hardwaresoftware)over the last decade (incl Turing award for Clarke)Widely used in industry Intel Microsoft IBM etc
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
69
Satisfiability (SATCSP) or Inference
Really solving the same problem but SATCSP view appears more effective
iff M(KB) micro M(reg) iff (KB AElig reg) is unsat KB sup2 reg
Assume KB and reg is in CNF (clausal form) Note that KB sup2 (macr AElig deg) iff (KB sup2 macr) and (KB sup2 deg) (consider defn in terms of models)So can break conjunction in several queries one foreach disjunction (clause) Each a call to a SAT solver to checkKB AElig (l1 Ccedil l2 hellip ) for consist or equivalently check KB AElig l1 AElig l2 hellip (ie CNF form) for consistency
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
70
KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip The SAT solvers essentially views the KB as a set of constraints that
defines a subset of the set of all 2N possible worlds (N prop vars)
The query reg also defines a subset of the 2N possible worlds (Above
we used a single clause (l1 Ccedil l2 hellip) where li is a var or its negation)
The SATCSP solvers figures out whether there are any worlds
consistent with the KB constraints that do not satisfy the constraints
of the query If so than the query does not follow from the KB
Aside In verification such a world exposes a bug
If such worlds do not exist then the query must be entailed by the KB
(M(KB) micro M(reg) Search starts from query)
Viewing logical reasoning as reasoning about constraints on the
way the ldquoworld can berdquo is quite actually quite natural Itrsquos definitely
a computationally effective strategy
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
71
Some examples of SAT solving
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Application I --- Diagnosis
bull Problem diagnosis a malfunctioning devicendash Carndash Computer systemndash Spacecraft
bull wherendash Design of the device is knownndash We can observe the state of only certain parts of
the device ndash much is hidden
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Model-Based Consistency-Based Diagnosis
bull Idea create a logical formula that describes how the device should workndash Associated with each ldquobreakablerdquo component C is a
proposition that states ldquoC is okayrdquondash Sub-formulas about component C are all conditioned on
C being okaybull A diagnosis is a smallest of ldquonot okayrdquo assumptions
that are consistent with what is actually observed
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Consistency-Based Diagnosis
1 Make some Observations O2 Initialize the Assumption Set A to assert that
all components are working properly3 Check if the KB A O together are inconsistent
(can deduce false)4 If so delete propositions from A until
consistency is restored (cannot deduce false) The deleted propositions are a diagnosis
There may be many possible diagnoses
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Example Automobile Diagnosisbull Observable Propositions
EngineRuns GasInTank ClockRunsbull Assumable Propositions
FuelLineOK BatteryOK CablesOK ClockOKbull Hidden (non-Assumable) Propositions
GasInEngine PowerToPlugsbull Device Description F
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns(BatteryOK CablesOK) PowerToPlugs(BatteryOK ClockOK) ClockRuns
bull Observations EngineRuns GasInTank ClockRuns
Note of course a highly simplified set of axioms
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Example
bull Is F Observations Assumptions consistent
bull F EngineRuns GasInTank ClockRuns FuelLineOK BatteryOK CablesOK ClockOK falsendash Must restore consistency
bull F EngineRuns GasInTank ClockRuns BatteryOK CablesOK ClockOK satisfiablendash FuelLineOK is a diagnosis
bull F EngineRuns GasInTank ClockRuns FuelLineOK CablesOK ClockOK falsendash BatteryOK is not a diagnosis
(GasInTank FuelLineOK) GasInEngine(GasInEngine PowerToPlugs) EngineRuns
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Complexity of Diagnosisbull If F is Horn then each consistency test takes linear O(n)
time ndash unit propagation is complete for Horn clausesbull Complexity = ways to delete propositions from
Assumption Set that are consideredndash Single fault diagnosis ndash O(n2)ndash Double fault diagnosis ndash O(n3)ndash Triple fault diagnosis ndash O(n4)
In practice for non-Horn use SAT solver for consistency check
Horn clause at most one positive literal Also consider )
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
NASA Deep Space Onebull Autonomous diagnosis amp repair ldquoRemote
Agentrdquobull Compiled systems schematic to 7000 var
SAT problem
Started January 1996Launch October 15th 1998Experiment May 17-21
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Deep Space One
bull a failed electronics unitndash Remote Agent fixed by reactivating the unit
bull a failed sensor providing false informationndash Remote Agent recognized as unreliable and therefore correctly ignored
bull an altitude control thruster (a small engine for controlling the spacecrafts orientation) stuck in the off position ndash Remote Agent detected and compensated for by switching to a mode that
did not rely on that thruster
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
II --- Testing Circuit Equivalence
bull Do two circuits compute the same function
bull Circuit optimizationbull Is there input for which
the two circuits compute different values
(Satisfying assignment will tell us Reveals bug)
A B
nand
A B
C Crsquo
Formal spec OR Possible implementationusing hardware gates (invertor amp nand)
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
A B
nand
A B
C Crsquo
D E
( )
( )
C A B
C D E
D A
E B
C C
Informally Given the same inputs values for A and Blogic specifies outputs (C and Crsquo) Are there inputs for whichC and Crsquo differ
Descr of OR
Descr ofHardware
Our queryDoes this holdYes iff negation withKB is inconsistent
Note if consitentmodel reveals ldquobugrdquoin hardware design
What happens if you addinstead of Still OK
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
III --- SAT Translation of N-Queens
bull At least one queen each row(Q11 v Q12 v Q13 v v Q18)(Q21 v Q22 v Q23 v v Q28)
bull No attacks (columns)(~Q11 v ~Q21)(~Q11 v ~Q31)(~Q11 v ~Q41)
bull No attacks (diag need left and right)(~Q11 v ~Q22)(~Q11 v ~Q33)(~Q11 v ~Q44)
How about No attacks (rows)(~Q11 v ~Q12)(~Q11 v ~Q13)(~Q11 v ~Q14)
hellipRedundant Why Sometimes slows solver
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
More compactlybull At least one queen each row(Qi1 v Qi2 v QiN) for 1 lt= i lt= N
bull No attacks (columns ldquolook in ith columnrdquo)(~Qji v ~Qjrsquoi) for 1 lt= ijjrsquo lt= N and j == jrsquo
bull No attacks (diag need left and right)(~Qij v ~Qirsquojrsquo) for 1 lt= i irsquo j jrsquo st |i ndash irsquo| = |j ndash jrsquo| amp i == irsquo amp j == jrsquo
Or in first order logic syntax eg second constraint set8 i 8 j 8 jrsquo (j == jrsquo ) ( Qji Ccedil Qjrsquoi)) with bounded type quantifier 1 lt= i j jrsquo lt= NReally a compact ldquopropositional schemardquoFirst-order logic for finite domains is equiv to prop logicFor SAT solver always ldquoground to propositionalrdquo
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
IV --- SAT Translation of Graph ColoringAt least one color per node i
(C_i1 v Q_i2 v Q_i3 v v Q_iK)
At most one color per node i (~C_ik v ~Q_ikrsquo) for all k == krsquo
If node i and node j (== i) share an edge need to have different colors
(~C_il v ~Q_jl) for all 1 lt= I lt= K
C_ik for node i has color kTotal colors K Total nodes N
Note Translation fromldquoproblemrdquo into SATReverse of usualtranslation to show NP-completeness
Works also for (easy)polytime problems
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
V --- Symbolic Model Checkingbull Any finite state machine is characterized by a transition function
ndash CPUndash Networking protocol
bull Wish to prove some invariant holds for any possible inputsbull Bounded model checking formula is sat iff invariant fails k steps
in the future
1
10
vector of Booleans representing
state of machine at time
01
(
t
k
i i i o ki
S
t
State Input State
State
S S I S S
1) Can go to CNF2) Equivalent to planning as propositional inferenceSATPLAN(Kautz amp Selman 1996)
774 RampNThe k-step planis satisfying assignment Note frac12 is just like our ldquomoverdquo earlier
Axioms says what happens ldquonextrdquo
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
A real-world example
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
ie ((not x1) or x7) and ((not x1) or x6)
and hellip etc
Bounded Model Checking instance
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
(x177 or x169 or x161 or x153 hellip or x17 or x9 or x1 or (not x185))
clauses constraints are getting more interestinghellip
10 pages later
hellip
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
4000 pages later
hellip
a 59-cnf clausehellip
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Finally 15000 pages later
The Chaff SAT solver (Princeton) solves this instance in less than one minute
Note that hellip
What makes this possible
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Progress in Last 20 yearsbull Significant progress since the 1990rsquos How muchbull Problem size We went from 100 variables 200 constraints (early 90rsquos)
to 1000000+ variables and 5000000+ constraints in 20 years
bull Search space from 10^30 to 10^300000[Aside ldquoone can encode quite a bit in 1M variablesrdquo]
bull Is this just Moorersquos Law It helped but not muchhellip ndash 2x faster computers does not mean can solve 2x larger instances ndash search difficulty does not scale linearly with problem size In fact for O(2^n) 2x faster handle 1 more variable Mainly algorithmic progress Memory growth also key
bull Tools 50+ competitive SAT solvers available (eg Minisat solver)bull See httpwwwsatcompetitionorg
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-
Forces Driving Faster Better SAT SolversInference engines
bull From academically interesting to practically relevant ldquoRealrdquo benchmarks with real interest in solving them
bull Regular SAT Solver Competitions (Germany-89 Dimacs-93 China-96 SAT-02 SAT-03 hellip SAT-07 SAT-09 SAT-2011)ndash ldquoIndustrial-instances-onlyrdquo SAT Races (2008 2010)ndash A tremendous resource Eg SAT Competition 2006 (Seattle)
bull 35+ solvers submitted downloadable mostly open sourcebull 500+ industrial benchmarks 1000+ other benchmarksbull 50000+ benchmark instances available on the Internet
bull Constant improvement in SAT solvers is the key to the success of eg SAT-based planning verification and KB inference
- CS 4700 Foundations of Artificial Intelligence
- Slide 2
- Knowledge and Reasoning
- Slide 4
- Knowledge-base Agents
- Knowledge bases
- A simple knowledge-based agent
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Intermezzo Reflections on Inference Knowledge and Data ca
- Watson
- Slide 28
- Watson (2)
- Semantic Web
- AI Knowledge- Data- Inference Triangle 2012
- Back to Knowledge-Based Agentshellip
- Illustrative example Wumpus World
- Wumpus world characterization
- Exploring a wumpus world
- Slide 37
- Further exploration
- Slide 39
- Slide 40
- Formally Entailment
- Slide 42
- Entailment in Wumpus World
- Wumpus models
- Entailment via ldquoModel Checkingrdquo
- Example redux More formal
- Wumpus World KB
- What about Time What about Actions
- Slide 49
- Successor-State Axioms
- Some example inferences Section 771 RampN
- Alternative formulation Situation Calculus RampN 1042
- Inference by enumeration ldquomodel checkingrdquo Style I
- All equivalent Prop FO Logic
- Proof techniques
- Aside
- Slide 57
- Slide 58
- Style II Proof by inference rules Modus Ponens (MP)
- Length of Proofs
- Style III Resolution
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Length of Proofs (2)
- Style IV SAT Solvers
- DPLL improvements
- Satisfiability (SATCSP) or Inference
- KB sup2 (l1 Ccedil l2 hellip) iff KB AElig l1 AElig l2 hellip
- Some examples of SAT solving
- Application I --- Diagnosis
- Model-Based Consistency-Based Diagnosis
- Consistency-Based Diagnosis
- Example Automobile Diagnosis
- Example
- Complexity of Diagnosis
- NASA Deep Space One
- Deep Space One
- II --- Testing Circuit Equivalence
- Slide 81
- III --- SAT Translation of N-Queens
- More compactly
- IV --- SAT Translation of Graph Coloring
- V --- Symbolic Model Checking
- A real-world example
- Slide 87
- Slide 88
- Slide 89
- Slide 90
- Progress in Last 20 years
- Forces Driving Faster Better SAT Solvers Inference engines
-