ITCS 3153Artificial Intelligence

Lecture 11Lecture 11

Logical AgentsLogical Agents

Chapter 7Chapter 7

Lecture 11Lecture 11

Logical AgentsLogical Agents

Chapter 7Chapter 7

Reasoning w/ propositional logic

Remember what we’ve developed so farRemember what we’ve developed so far

• Logical sentencesLogical sentences

• And, or, not, implies (entailment), iff (equivalence)And, or, not, implies (entailment), iff (equivalence)

• Syntax vs. semanticsSyntax vs. semantics

• Truth tablesTruth tables

• Satisfiability, proof by contradictionSatisfiability, proof by contradiction

Remember what we’ve developed so farRemember what we’ve developed so far

• Logical sentencesLogical sentences

• And, or, not, implies (entailment), iff (equivalence)And, or, not, implies (entailment), iff (equivalence)

• Syntax vs. semanticsSyntax vs. semantics

• Truth tablesTruth tables

• Satisfiability, proof by contradictionSatisfiability, proof by contradiction

Logical Equivalences

Know these equivalencesKnow these equivalencesKnow these equivalencesKnow these equivalences

Reasoning w/ propositional logic

Inference RulesInference Rules

• Modus Ponens: Modus Ponens:

– Whenever sentences of form Whenever sentences of form => => and and are given are giventhe sentence the sentence can be inferred can be inferred

RR11: Green => Martian: Green => Martian

RR22: Green: Green

Inferred: MartianInferred: Martian

Inference RulesInference Rules

• Modus Ponens: Modus Ponens:

– Whenever sentences of form Whenever sentences of form => => and and are given are giventhe sentence the sentence can be inferred can be inferred

RR11: Green => Martian: Green => Martian

RR22: Green: Green

Inferred: MartianInferred: Martian

Reasoning w/ propositional logic

Inference RulesInference Rules• And-EliminationAnd-Elimination

– Any of conjuncts can be inferredAny of conjuncts can be inferred

RR11: Martian ^ Green: Martian ^ Green

Inferred: MartianInferred: Martian

Inferrred: GreenInferrred: Green

Use truth tables if you want to confirm inference Use truth tables if you want to confirm inference rulesrules

Inference RulesInference Rules• And-EliminationAnd-Elimination

– Any of conjuncts can be inferredAny of conjuncts can be inferred

RR11: Martian ^ Green: Martian ^ Green

Inferred: MartianInferred: Martian

Inferrred: GreenInferrred: Green

Use truth tables if you want to confirm inference Use truth tables if you want to confirm inference rulesrules

Example of a proof

~P ~B

BP?

P?

P?

P?

Example of a proof

~P ~B

B~P

P?

P?

~P

Constructing a proof

Proving Proving is like is like searchingsearching

• Find sequence of logical inference rules that lead to desired Find sequence of logical inference rules that lead to desired resultresult

• Note the explosion of propositionsNote the explosion of propositions

– Good proof methods ignore the countless irrelevant Good proof methods ignore the countless irrelevant propositionspropositions

Proving Proving is like is like searchingsearching

• Find sequence of logical inference rules that lead to desired Find sequence of logical inference rules that lead to desired resultresult

• Note the explosion of propositionsNote the explosion of propositions

– Good proof methods ignore the countless irrelevant Good proof methods ignore the countless irrelevant propositionspropositions

Monotonicity of knowledge base

Knowledge base can only get largerKnowledge base can only get larger

• Adding new sentences to knowledge base can only make it get largerAdding new sentences to knowledge base can only make it get larger

– If (KB entails If (KB entails ))

((KB ^ ((KB ^ ) entails ) entails ))

• This is important when constructing proofsThis is important when constructing proofs

– A logical conclusion drawn at one point cannot be invalidated by a A logical conclusion drawn at one point cannot be invalidated by a subsequent entailmentsubsequent entailment

Knowledge base can only get largerKnowledge base can only get larger

• Adding new sentences to knowledge base can only make it get largerAdding new sentences to knowledge base can only make it get larger

– If (KB entails If (KB entails ))

((KB ^ ((KB ^ ) entails ) entails ))

• This is important when constructing proofsThis is important when constructing proofs

– A logical conclusion drawn at one point cannot be invalidated by a A logical conclusion drawn at one point cannot be invalidated by a subsequent entailmentsubsequent entailment

How many inferences?

Previous example relied on application of inference Previous example relied on application of inference rules to generate new sentencesrules to generate new sentences

• When have you drawn enough inferences to prove something?When have you drawn enough inferences to prove something?

– Too many make search process take longerToo many make search process take longer

– Too few halt logical progression and make proof process Too few halt logical progression and make proof process incompleteincomplete

Previous example relied on application of inference Previous example relied on application of inference rules to generate new sentencesrules to generate new sentences

• When have you drawn enough inferences to prove something?When have you drawn enough inferences to prove something?

– Too many make search process take longerToo many make search process take longer

– Too few halt logical progression and make proof process Too few halt logical progression and make proof process incompleteincomplete

Resolution

Unit Resolution Inference RuleUnit Resolution Inference Rule

• If If mm and and llii are are

complementarycomplementaryliteralsliterals

Unit Resolution Inference RuleUnit Resolution Inference Rule

• If If mm and and llii are are

complementarycomplementaryliteralsliterals

Resolution Inference Rule

Also works with clausesAlso works with clauses

But make sure each literal appears only onceBut make sure each literal appears only once

Also works with clausesAlso works with clauses

But make sure each literal appears only onceBut make sure each literal appears only once

Resolution and completeness

Any Any completecomplete search algorithm, applying only the search algorithm, applying only the resolution rule, can derive any conclusion resolution rule, can derive any conclusion entailed by any knowledge base in propositional entailed by any knowledge base in propositional logiclogic

• More specifically, More specifically, refutation completenessrefutation completeness

– Able to confirm or refute any sentenceAble to confirm or refute any sentence

– Unable to enumerate all true sentencesUnable to enumerate all true sentences

Any Any completecomplete search algorithm, applying only the search algorithm, applying only the resolution rule, can derive any conclusion resolution rule, can derive any conclusion entailed by any knowledge base in propositional entailed by any knowledge base in propositional logiclogic

• More specifically, More specifically, refutation completenessrefutation completeness

– Able to confirm or refute any sentenceAble to confirm or refute any sentence

– Unable to enumerate all true sentencesUnable to enumerate all true sentences

What about “and” clauses?

Resolution only applies to “or” clausesResolution only applies to “or” clauses

• Every sentence of propositional logic can be transformed to a Every sentence of propositional logic can be transformed to a logically equivalent conjunction of disjunctions of literalslogically equivalent conjunction of disjunctions of literals

Conjunctive Normal Form (CNF)Conjunctive Normal Form (CNF)

• A sentence expressed as conjunction of disjunction of literalsA sentence expressed as conjunction of disjunction of literals

– k-CNF: exactly k literals per clausek-CNF: exactly k literals per clause

Resolution only applies to “or” clausesResolution only applies to “or” clauses

• Every sentence of propositional logic can be transformed to a Every sentence of propositional logic can be transformed to a logically equivalent conjunction of disjunctions of literalslogically equivalent conjunction of disjunctions of literals

Conjunctive Normal Form (CNF)Conjunctive Normal Form (CNF)

• A sentence expressed as conjunction of disjunction of literalsA sentence expressed as conjunction of disjunction of literals

– k-CNF: exactly k literals per clausek-CNF: exactly k literals per clause

CNF

An algorithm for resolution

We wish to prove KB entails We wish to prove KB entails • Must show (KB ^ ~Must show (KB ^ ~) is unsatisfiable) is unsatisfiable

– No possible way for KB to entail (not No possible way for KB to entail (not ))

– Proof by contradictionProof by contradiction

We wish to prove KB entails We wish to prove KB entails • Must show (KB ^ ~Must show (KB ^ ~) is unsatisfiable) is unsatisfiable

– No possible way for KB to entail (not No possible way for KB to entail (not ))

– Proof by contradictionProof by contradiction

An algorithm for resolution

AlgorithmAlgorithm

• KB ^ ~KB ^ ~is put in CNFis put in CNF

• Each pair with complementary literals is resolved to produce Each pair with complementary literals is resolved to produce new clause which is added to KB (if novel)new clause which is added to KB (if novel)

– Cease if no new clauses to add (~Cease if no new clauses to add (~ is not entailed) is not entailed)

– Cease if resolution rule derives empty clause (~Cease if resolution rule derives empty clause (~ is is entailed)entailed)

AlgorithmAlgorithm

• KB ^ ~KB ^ ~is put in CNFis put in CNF

• Each pair with complementary literals is resolved to produce Each pair with complementary literals is resolved to produce new clause which is added to KB (if novel)new clause which is added to KB (if novel)

– Cease if no new clauses to add (~Cease if no new clauses to add (~ is not entailed) is not entailed)

– Cease if resolution rule derives empty clause (~Cease if resolution rule derives empty clause (~ is is entailed)entailed)

Example of resolution

Proof that there is not a pit in PProof that there is not a pit in P1,21,2: ~P: ~P1,21,2

• KB ^ PKB ^ P1,21,2 leads to empty clause leads to empty clause

• Therefore ~PTherefore ~P1,21,2 is true is true

Proof that there is not a pit in PProof that there is not a pit in P1,21,2: ~P: ~P1,21,2

• KB ^ PKB ^ P1,21,2 leads to empty clause leads to empty clause

• Therefore ~PTherefore ~P1,21,2 is true is true

Formal Algorithm

Horn Clauses

Horn ClauseHorn Clause

• Disjunction of literals with at most one is positiveDisjunction of literals with at most one is positive

– (~a V ~b V ~c V d)(~a V ~b V ~c V d)

– (~a V b V c V ~d) (~a V b V c V ~d) Not a Horn ClauseNot a Horn Clause

Horn ClauseHorn Clause

• Disjunction of literals with at most one is positiveDisjunction of literals with at most one is positive

– (~a V ~b V ~c V d)(~a V ~b V ~c V d)

– (~a V b V c V ~d) (~a V b V c V ~d) Not a Horn ClauseNot a Horn Clause

Horn Clauses

Can be written as a special implicationCan be written as a special implication

• (~a V ~b V c) (~a V ~b V c) (a ^ b) => c (a ^ b) => c

– (~a V ~b V c) == (~(a ^ b) V c) … de Morgan(~a V ~b V c) == (~(a ^ b) V c) … de Morgan

– (~(a ^ b) V c) == ((a ^ b) => c … implication elimination(~(a ^ b) V c) == ((a ^ b) => c … implication elimination

Can be written as a special implicationCan be written as a special implication

• (~a V ~b V c) (~a V ~b V c) (a ^ b) => c (a ^ b) => c

– (~a V ~b V c) == (~(a ^ b) V c) … de Morgan(~a V ~b V c) == (~(a ^ b) V c) … de Morgan

– (~(a ^ b) V c) == ((a ^ b) => c … implication elimination(~(a ^ b) V c) == ((a ^ b) => c … implication elimination

Horn Clauses

Permit straightforward inference determinationPermit straightforward inference determination

• Forward chainingForward chaining

• Backward chainingBackward chaining

Permit straightforward inference determinationPermit straightforward inference determination

• Forward chainingForward chaining

• Backward chainingBackward chaining

Horn Clauses

Permit determination of entailment in linear time Permit determination of entailment in linear time (in order of knowledge base size)(in order of knowledge base size)Permit determination of entailment in linear time Permit determination of entailment in linear time (in order of knowledge base size)(in order of knowledge base size)

Forward Chaining

Forward Chaining

PropertiesProperties

• SoundSound

• CompleteComplete

– All entailed atomic sentence will be derivedAll entailed atomic sentence will be derived

Data DrivenData Driven

• Start with what we knowStart with what we know

• Derive new info until we discover what we wantDerive new info until we discover what we want

PropertiesProperties

• SoundSound

• CompleteComplete

– All entailed atomic sentence will be derivedAll entailed atomic sentence will be derived

Data DrivenData Driven

• Start with what we knowStart with what we know

• Derive new info until we discover what we wantDerive new info until we discover what we want

Backward Chaining

Start with what you want to know, a query (q)Start with what you want to know, a query (q)

Look for implications that conclude qLook for implications that conclude q

Goal-Directed ReasoningGoal-Directed Reasoning

Start with what you want to know, a query (q)Start with what you want to know, a query (q)

Look for implications that conclude qLook for implications that conclude q

Goal-Directed ReasoningGoal-Directed Reasoning