constraint satisfaction problem (answer/do something) 4 10/30/2017 3 agents with knowledge and...

10/30/2017

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Logical AgentsDeepak Kumar

October-November 2017

What is AI? – Recall from Week 1 & 2

Reasoning with Logic

• Aristotle: What are correct arguments/thought processes?

• Formal Logics:

Socrates is human.All humans are mortal.Therefore Socrates is mortal.

• Laws of thought govern the operation of the mind.

Machines with logic“laws of thought”

Logic

Reasoning Etc.

Thinking Rationally

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Computer Science - Databases

• Database Systems

Database

Data

Query

Response

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Agents with Knowledge and Reasoning

• Knowledge-Based Systems

KnowledgeBase

“Knowledge”

Tell/Sense/Ask

Response (Answer/Do Something)

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• Knowledge-Based Systems

• Knowledge – set of sentences that describe facts about the world (or domain)

KnowledgeBase

“Knowledge”

Tell/Sense/Ask


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• Knowledge Representation & Reasoning (KRR) Systems

• Knowledge – set of sentences that describe facts about the world (or domain)

• Inferences – procedures/rules that operate on facts to infer new facts

KnowledgeBase

“Knowledge”+

Inference Engine

Tell/Sense/Ask


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Examples – Rule-Based Systems

• MYCIN

if stain of organism is gram positive, andmorphology of organism is coccus, andthe growth confirmation of organism is chains

thenid of organism is streptococcus

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Examples – KRR Systems

• BLOCKSWORLD with declarative knowledge

if a block is on top of another blockthen

the latter block is not clear

Facts

blue block is on top of red blockred block is on the tableblue block is clear

∴ red block is not clear

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Knowledge Representation

• Syntax: How sentences are formed

• Semantics: Meaning of sentences

• Computation: How sentences are manipulated

Facts


∴ red block is not clear

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red block is not clear

KR Sentences KR Sentences

Facts about the world Fact about the world

inference

follows

World

Representation

sem

anti

cs

sem

anti

cs


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red block is not clear

KR Sentences KR Sentences


inference

follows

World

Representation

sem

anti

cs

sem

anti

cs


Logic is one way of doing this!11

What is a logic?

• Study of correct inferences

Premises

ቋ

………

True

Conclusion

∴ ሽ… is also True

If it is cold then my car will not start.My car will not start.

∴ It is cold.

If it is cold then my car will not start.It is cold.

∴ My car will not start.

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What is a logic?

• Study of correct inferences – Truth preserving consequences

Premises

ቋ

………

True

Conclusion



∴ It is cold.



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What is a logic?


Premises

ቋ

………

True

Conclusion



∴ It is cold.



This is NOTTruth preserving.

This isTruth preserving.

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What is a logic?


Premises

ቋ

………

True

Conclusion



∴ It is cold.



This is NOTTruth preserving.This is sometimesused in diagnosis!

This isTruth preserving.

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What is a logic?

• Study of correct inferences

• Formalize the notion of correct inference

• Step1: Define a formal language to write sentences – syntaxwell-formed sentences (wffs)

• Step 2: What do the wffs mean? – Semantics/Model TheoryNeed an interpretation for wffs

• Step 3: Rules of Inference – Proof Theory

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Rules of Inference red block is not clearKR Sentences KR Sentences


inference

follows

World

Representation

sem

anti

cs

sem

anti

cs


Properties

Soundness: Every new sentence thatcan be derived from KR is avalid consequence.

Completeness: Every valid consequence ofKR can be formally derived.

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Rules of Inference red block is not clearKR Sentences KR Sentences


inference

follows

World

Representationse

man

tics

sem

anti

cs


Properties

Soundness: Every new sentence thatcan be derived from KR is avalid consequence.

Completeness: Every valid consequence ofKR can be formally derived.

This is easy to show

This is true for some logics.

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There are many types of logic

• Propositional Logic

• First-Order Logic

• Second-Oder Logic

• Temporal Logic

• Modal Logic

• Constraint Logic

• Etc.

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There are many types of logic

• Propositional Logic

• First-Order Logic

• Second-Oder Logic

• Temporal Logic

• Modal Logic

• Constraint Logic

• Etc.

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Propositional Logic

• A language for symbolic reasoning

• Proposition – a statement that is either True or False.

E.g.

Bryn Mawr College is located in Canada.Today is a sunny day.23 is a prime number.This class is boring.

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Propositional Logic - Syntax• Symbols

• Atoms:• Constants - 𝑇𝑟𝑢𝑒 (𝑇), 𝐹𝑎𝑙𝑠𝑒 (𝐹)• Propositional Symbols – 𝑃, 𝑄, 𝑅, 𝑅𝐴𝐼𝑁𝐼𝑁𝐺, 𝐶𝐿𝑂𝑈𝐷𝑌, 𝐵𝑂𝑅𝐼𝑁𝐺, etc.

• Connective Symbols - ∨, ∧, ¬, ⇒

• Sentences – Well-formed formulas (wffs)• Any atom is a wff [Atomic Sentences]

e.g. 𝑃, 𝑄, 𝑅, 𝑅3• Complex Sentences

• If ω1and ω2 are wffs, then so areω1ω1∨ ω2 disjunctionω1∧ ω2 conjunctionω1⇒ ω2 implication¬ ω1 negation

• There are no other wffs.22

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Propositional Logic - Syntax• Sentences – Well-formed formulas (wffs)

• Any atom is a wff [Atomic Sentences]e.g. 𝑃, 𝑄, 𝑅, 𝑅3

• Complex Sentences• If ω1and ω2 are wffs, then so are

ω1ω1∨ ω2 disjunctionω1∧ ω2 conjunctionω1⇒ ω2 implication¬ ω1 negation

• There are no other wffs.

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Examples

𝑃(𝑃 ∨ 𝑄) ⇒ ¬P𝑃 ∧ 𝑄¬ ¬ PP ⇒⇒ ¬P

etc.

Propositional Logic - Syntax• Sentences – Well-formed formulas (wffs)

• Any atom is a wff [Atomic Sentences]e.g. 𝑃, 𝑄, 𝑅, 𝑅3

• Complex Sentences• If ω1and ω2 are wffs, then so are

ω1ω1∨ ω2 disjunctionω1∧ ω2 conjunctionω1⇒ ω2 implication¬ ω1 negation

• There are no other wffs.

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Examples

𝑃(𝑃 ∨ 𝑄) ⇒ ¬P𝑃 ∧ 𝑄¬ ¬ P

P ⇒⇒ ¬P X

etc.

constraint satisfaction problem (answer/do something) 4 10/30/2017 3 agents with knowledge and...

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