Propositional LogicPropositional Logic

Russell and Norvig: Chapter 6Chapter 7, Sections 7.1—7.4Slides adapted from:robotics.stanford.edu/~latombe/cs121/2003/home.htm

Knowledge-Based AgentKnowledge-Based Agent

environmentagent

?

sensors

actuators

Knowledge base

Types of KnowledgeTypes of Knowledge

Procedural, e.g.: functions Such knowledge can only be used in one way -- by executing it

Declarative, e.g.: constraints It can be used to perform many different sorts of inferences

LogicLogic

Logic is a declarative language to:

Assert sentences representing facts that hold in a world W (these sentences are given the value true)

Deduce the true/false values to sentences representing other aspects of W

Connection World-Connection World-

RepresentationRepresentation

World W

Conceptualization

Facts about W

hold

hold

Sentences

represent

Facts about W

represent

Sentencesentail

Examples of LogicsExamples of Logics

Propositional calculus A B C First-order predicate calculus ( x)( y) Mother(y,x) Logic of Belief B(John,Father(Zeus,Cronus))

ModelModel

Assignment of a truth value – true or false – to every atomic sentenceExamples: Let A, B, C, and D be the propositional

symbols m = {A=true, B=false, C=false, D=true} is a

model m’ = {A=true, B=false, C=false} is not a

model

With n propositional symbols, one can define 2n models

Model of a KBModel of a KB

Let KB be a set of sentences

A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m Given a vocabulary A, B, C and D, how

many models for A^B -> C are there? for A^B -> B?

Satisfiability of a KBSatisfiability of a KB

A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable

KB1 = {P, QR} is satisfiable

KB2 = {PP} is satisfiable

KB3 = {P, P} is unsatisfiable

valid sentenceor tautology

Logical EntailmentLogical Entailment

KB : set of sentences : arbitrary sentence KB entails – written KB – iff every model of KB is also a model of Alternatively, KB iff {KB,} is unsatisfiable KB is valid

Inference RuleInference Rule

An inference rule {, } consists of 2 sentence patterns and called the conditions and one sentence pattern called the conclusion If and match two sentences of KB then the corresponding can be inferred according to the rule

InferenceInference

I: Set of inference rules KB: Set of sentences Inference is the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB

Example: Modus PonensExample: Modus Ponens

Battery-OK Bulbs-OK Headlights-WorkBattery-OK Starter-OK Empty-Gas-Tank Engine-StartsEngine-Starts Flat-Tire Car-OKBattery-OK Bulbs-OK

{ , }

{, }

Connective symbol (implication)

Logical entailment

Inference

KB iff KB is valid

SoundnessSoundness

An inference rule is sound if it generates only entailed sentences All inference rules previously given are sound, e.g.:modus ponens: { , } The following rule: { , } is unsound, which does not mean it is useless

CompletenessCompleteness

A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rulesModus ponens alone is not complete, e.g.:from A B and B, we cannot get A

ProofProof

The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

ProofProof

The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Battery-OK Starter-OK (5+6)10. Battery-OK Starter-OK Empty-Gas-Tank (9+7)11. Engine-Starts (2+10)12. Engine-Starts Flat-Tire (3+8)13. Flat-Tire (11+12)

Inference ProblemInference Problem

Given: KB: a set of sentence : a sentence

Answer: KB ?

KB iff {KB,} is unsatisfiable

Deduction vs. Satisfiability Deduction vs. Satisfiability TestTest

Hence:• Deciding whether a set of

sentences entails another sentence, or not

• Testing whether a set of sentences is satisfiable, or not

are closely related problems

Complementary LiteralsComplementary Literals

A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P or P

Two literals are complementary if one is the negation of the other, e.g.: P and P

Unit Resolution RuleUnit Resolution Rule

Given two sentences: L1 … Lp and M where Li,…, Lp and M are all literals, and M and Li are complementary literalsInfer: L1 … Li-1 Li+1 … Lp

ExamplesExamplesFrom:

Engine-Starts Car-OKEngine-Starts

Infer:Car-OK From:

Engine-Starts Car-OKCar-OK

Infer: Engine-Starts

Modus ponens

Modus tolens

Engine-Starts Car-OK

Shortcoming of Unit Shortcoming of Unit ResolutionResolution

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Empty-Gas-Tank

we can infer nothing!

Full Resolution RuleFull Resolution Rule

Given two sentences: L1 … Lp and M1 … Mq where L1,…, Lp, M1,…, Mq are all literals, and Li and Mj are complementary literalsInfer: L1 … Li-1Li+1…LkM1 … Mj-1Mj+1…Mk

in which only one copy of each literal is retained (factoring)

ExampleExample

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Empty-Gas-Tank

Infer:

Empty-Gas-Tank Flat-Tire Car-OK

ExampleExample

From:

P Q ( P Q)

Q R ( Q R)

Infer:

P R ( P R)

Not All Inferences are Not All Inferences are Useful! Useful!

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Flat-Tire

Infer:

Flat-Tire Flat-Tire Car-OK

Not All Inferences are Not All Inferences are Useful!Useful!

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Flat-Tire

Infer:

Flat-Tire Flat-Tire Car-OK

tautology

Not All Inferences are Not All Inferences are Useful!Useful!

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Flat-Tire

Infer:

Flat-Tire Flat-Tire Car-OK True

tautology

Full Resolution RuleFull Resolution Rule

Given two clauses: L1 … Lp and M1 … Mq

Infer the clause: L1 … Li-1Li+1…LkM1 … Mj-1Mj+1…Mk

Sentence Sentence Clause Form Clause FormExample:

(A B) (C D)

1. Eliminate (A B) (C D)2. Reduce scope of (A B) (C D)3. Distribute over

(A (C D)) (B (C D))(A C) (A D) (B C) (B D)

Set of clauses:{A C , A D , B C , B D}

Resolution Refutation Resolution Refutation AlgorithmAlgorithm

RESOLUTION-REFUTATION(KB)clauses set of clauses obtained from KB and new {}Repeat:

For each C, C’ in clauses dores RESOLVE(C,C’)If res contains the empty clause then

return yesnew new U res

If new clauses then return noclauses clauses U new

ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Flat-Tire

SummarySummary

Propositional Logic Model of a KB Logical entailment Inference rules Resolution rule Clause form of a set of sentences Resolution refutation algorithm