first-order logic.milos/courses/cs2710-fall...– an interpretation gives a semantic to primitives....

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CS 2710 Foundations of AI

CS 2710 Foundations of AILecture 11

Milos [email protected] Sennott Square

First-order logic.


Administration

• PS-4:– Due on Thursday, October 7, 2004

• Midterm: – October 19, 2004 during the class– Closed book– Covers:

Search and Knowledge Representation

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LogicLogic is defined by:• A set of sentences

– A sentence is constructed from a set of primitives according to syntax rules.

• A set of interpretations– An interpretation gives a semantic to primitives. It associates

primitives with objects, values in the real world.• The valuation (meaning) function V

– Assigns a truth value to a given sentence under some interpretation

interpretasentence: ×V },{tion FalseTrue→


First-order logic. Syntax.

Term - syntactic entity for representing objects

Terms in FOL:• Constant symbols: represent specific objects

– E.g. John, France, car89• Variables: represent objects of a certain type (type = domain of

discourse)– E.g. x,y,z

• Functions applied to one or more terms– E.g. father-of (John)

father-of(father-of(John))

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First order logic. Syntax.

Sentences in FOL:• Atomic sentences:

– A predicate symbol applied to 0 or more termsExamples:Red(car12), Sister(Amy, Jane);Manager(father-of(John));

– t1 = t2 equivalence of termsExample:John = father-of(Peter)


First order logic. Syntax.

Sentences in FOL:• Complex sentences:• Assume are sentences in FOL. Then:

–and

–are sentences

Symbols- stand for the existential and the universal quantifier

)( ψφ ∧ )( ψφ ∨ ψ¬)( ψφ ⇔)( ψφ ⇒ψφ ,

φx∀ φy∃

∀∃,

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Semantics. Interpretation. An interpretation I is defined by a mapping of symbols to the

domain of discourse D or relations on D• domain of discourse: a set of objects in the world we represent

and refer to; An interpretation I maps:• Constant symbols to objects in D

I(John) = • Predicate symbols to relations, properties on D

• Function symbols to functional relations on D

; ; ….

; ; ….

I(brother) =

I(father-of) =


Semantics of sentences. Meaning (evaluation) function:

A predicate predicate(term-1, term-2, term-3, term-n) is true for the interpretation I , iff the objects referred to by term-1, term-2, term-3, term-n are in the relation referred to by predicate

interpretasentence: ×V },{tion FalseTrue→

V(brother(John, Paul), I) = True

; ; ….I(brother) =

I(John) = I(Paul) =

brother(John, Paul) = in I(brother)

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Semantics of sentences.

• Equality

• Boolean expressions: standard

• Quantifications

V(term-1= term-2, I) = TrueIff I(term-1) =I(term-2)

V(sentence-1 ∨ sentence-2, I) = TrueIff V(sentence-1,I)= True or V(sentence-2,I)= True

E.g.

V( , I) = TrueIff for all V( ,I[x/d])= Trueφx∀

Dd ∈ φ

V( , I) = TrueIff there is a , s.t. V( ,I[x/d])= Trueφx∃

Dd ∈ φ

substitution of x with d


Order of quantifiers

• Order of quantifiers of the same type does not matter

• Order of different quantifiers changes the meaning

For all x and y, if x is a parent of y then y is a child of x)x,(),(, ychildyxparentyx ⇒∀)x,(),(, ychildyxparentxy ⇒∀

),( yxlovesyx∃∀

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Everybody loves somebody


),( yxlovesyx∃∀

),( yxlovesxy∀∃





Everybody loves somebody

There is someone who is loved by everyone


),( yxlovesyx∃∀

),( yxlovesxy∀∃

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Connections between quantifiers

Everyone likes ice cream

Is it possible to convey the same meaning using an existential quantifier ?

A universal quantifier in the sentence can be expressed using anexistential quantifier !!!

),( IceCreamxlikesx∀

),( IceCreamxlikesx ¬¬∃There is no one who does not like ice cream


Connections between quantifiers

Is it possible to convey the same meaning using a universal quantifier ?

An existential quantifier in the sentence can be expressed using a universal quantifier !!!

Someone likes ice cream

),( IceCreamxlikesx∃

Not everybody does not like ice cream

),( IceCreamxlikesx ¬¬∀

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Representing knowledge in FOLExample:

Kinship domain

• Objects: people

• Properties: gender

• Relations: parenthood, brotherhood, marriage

• Functions: mother-of (one for each person x)

)(),( xFemalexMale

),(),,(),,( yxSpouseyxBrotheryxParent

)( xMotherOf

K,,, JaneMaryJohn


Kinship domain in FOL

Relations between predicates and functions: write down what we know about them; how relate to each other.

• Male and female are disjoint categories

• Parent and child relations are inverse

• A grandparent is a parent of parent

• A sibling is another child of one’s parents

• And so on ….

),(),(),(, cpParentpgParentpcgtGrandparencg ∧∃⇔∀

),(),()(),(, ypParentxpParentpyxyxSiblingyx ∧∃∧≠⇔∀

),(),(, xyChildyxParentyx ⇔∀

)()( xFemalexMalex ¬⇔∀

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Inference in First order logic


Logical inference in FOL

Logical inference problem:• Given a knowledge base KB (a set of sentences) and a

sentence , does the KB semantically entail ?

In other words: In all interpretations in which sentences in the KB are true, is also true?

Logical inference problem in the first-order logic is undecidable !!!. No procedure that can decide the entailment for all possible input sentences in a finite number of steps.

α=|KB ?

α

αα

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Logical inference problem in the Propositional logic

Computational procedures that answer:

Three approaches:• Truth-table approach• Inference rules• Conversion to the inverse SAT problem

– Resolution-refutation

α=|KB ?


Inference in FOL: Truth table approach

• Is the Truth-table approach a viable approach for the FOL?

• NO! • Why? • It would require us to enumerate and list all possible

interpretations I • I = (assignments of symbols to objects, predicates to

relations and functions to relational mappings)• Simply there are too many interpretations

?

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Inference in FOL: Inference rules

• Is the Inference rule approach a viable approach for the FOL?

• Yes. • The inference rules represent sound inference patterns one

can apply to sentences in the KB• What is derived follows from the KB• Caveat: we need to add rules for handling quantifiers

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Inference rules

• Inference rules from the propositional logic:– Modus ponens

– Resolution

– and others: And-introduction, And-elimination, Or-introduction, Negation elimination

• Additional inference rules are needed for sentences with quantifiers and variables– Must involve variable substitutions

BABA ,⇒

CACBBA

∨∨¬∨ ,

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Sentences with variablesFirst-order logic sentences can include variables.• Variable is:

– Bound – if it is in the scope of some quantifier

– Free – if it is not bound.

• Sentence (formula) is:– Closed – if it has no free variables

– Open – if it is not closed– Ground – if it does not have any variables

)(xPx∀

),( JaneJohnLikes

)()( xQyPx ∧∃ y is free

)()( xQyPxy ⇒∃∀


Variable substitutions

• Variables in the sentences can be substituted with terms.(terms = constants, variables, functions)

• Substitution:– Is represented by a mapping from variables to terms

– Application of the substitution to sentences

},/,/{ 2211 Ktxtx

),()),(},/,/({ PamSamLikesyxLikesPamySamxSUBST =

))(,()),()},(/,/({

JohnfatherofzLikesyxLikesJohnfatherofyzxSUBST =

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Inference rules for quantifiers

• Universal elimination

– substitutes a variable with a constant symbol

• Existential elimination.

– Substitutes a variable with a constant symbol that does not appear elsewhere in the KB

)()(

axx

φφ∃

)()(

axx

φφ∀

a - is a constant symbol

),( IceCreamxLikesx∀ ),( IceCreamBenLikes

),( VictimxKillx∃ ),( VictimMurdererKill


Inference rules for quantifiers

• Universal instantiation (introduction)

– Introduces a universal variable which does not affect or its assumptions

• Existential instantiation (introduction)

– Substitutes a ground term in the sentence with a variable and an existential statement

)()(xx

aφφ

∃

φφx∀

x – is not free in φ

a – is a ground term inx – is not free in

φφ

φ

),( IceCreamxLikesx∃),( IceCreamBenLikes

),( JaneAmySister ),( JaneAmySisterx∀

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Unification

• Problem in inference: Universal elimination gives many opportunities for substituting variables with ground terms

• Solution: Try substitutions that may help– Use substitutions of “similar” sentences in KB

• Unification – takes two similar sentences and computes the substitution that makes them look the same, if it exists

),(), s.t. ),( qSUBSTpSUBST( σqpUNIFY σσ ==

)()(

axx

φφ∀

a - is a constant symbol


Unification. Examples.

• Unification:

• Examples:

}/{)),(),,(( JanexJaneJohnKnowsxJohnKnowsUNIFY =

}/,/{)),(),,(( JohnyAnnxAnnyKnowsxJohnKnowsUNIFY =

failElizabethxKnowsxJohnKnowsUNIFY =)),(),,((

}/),(/{ )))(,(),,((

JohnyJohnMotherOfxyMotherOfyKnowsxJohnKnowsUNIFY

=

),(), s.t. ),( qSUBSTpSUBST( σqpUNIFY σσ ==

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Generalized inference rules.

• Use substitutions that let us make inferencesExample: Modus Ponens• If there exists a substitution such that

• Substitution that satisfies the generalized inference rule can be build via unification process

• Advantage of the generalized rules: they are focused– only substitutions that allow the inferences to proceed

),(',',', 2121

BSUBSTAAABAAA nn

σKK ⇒∧∧

)',(),( ii ASUBSTASUBST σσ =

σ

for all i=1,2, n


Resolution inference rule• Recall: Resolution inference rule is sound and complete

(refutation-complete) for the propositional logic and CNF

• Generalized resolution rule is sound and refutation completefor the first-order logic and CNF w/o equalities (if unsatisfiablethe resolution will find the contradiction)

CBCABA

∨∨¬∨ ,

),(,

111111

2121

njjkii

nk

SUBST ψψψψφφφφσψψψφφφ

KKKK

KK

+−+− ∨∨∨∨∨∨∨∨∨∨∨∨

failUNIFY ji ≠¬= ),( ψφσ

Example:)()(

)()(),()(ySJohnP

ySJohnQxQxP∨

∨¬∨

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Inference with resolution rule

• Proof by refutation:– Prove that is unsatisfiable– resolution is refutation-complete

• Main procedure (steps):1. Convert to CNF with ground terms and

universal variables only2. Apply repeatedly the resolution rule while keeping track

and consistency of substitutions3. Stop when empty set (contradiction) is derived or no more

new resolvents (conclusions) follow

α¬,KB

α¬,KB


Conversion to CNF

1. Eliminate implications, equivalences

2. Move negations inside (DeMorgan’s Laws, double negation)

3. Standardize variables (rename duplicate variables)

)()( qpqp ∨¬→⇒

qpqp ¬∨¬→∧¬ )(qpqp ¬∧¬→∨¬ )(

pxpx ¬∃→¬∀pxpx ¬∀→¬∃

pp →¬¬

))(())(())(())(( yQyxPxxQxxPx ∃∨∀→∃∨∀

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Conversion to CNF

4. Move all quantifiers left (no invalid capture possible )

5. Skolemization (removal of existential quantifiers through elimination)

• If no universal quantifier occurs before the existential quantifier, replace the variable with a new constant symbol

• If a universal quantifier precede the existential quantifier replace the variable with a function of the “universal” variable

)()())(())(( yQxPyxyQyxPx ∨∃∀→∃∨∀

))(()()()( xFQxPxyQxPyx ∨∀→∨∃∀

)()()()( BQAPyQAPy ∨→∨∃

)( xF - a Skolem function


Conversion to CNF

6. Drop universal quantifiers (all variables are universally quantified)

7. Convert to CNF using the distributive laws

The result is a CNF with variables, constants, functions

)()()( rpqprqp ∨∧∨→∧∨

))(()())(()( xFQxPxFQxPx ∨→∨∀

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Resolution example

)()( wQwP ∨¬ )()( ySyQ ∨¬ )()( zSzR ∨¬)()( xRxP ∨

KB

)(AS¬

α¬, , , ,


Resolution example


KB

)(AS¬

α¬

)()( wSwP ∨¬

, , , ,

}/{ wy

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Resolution example


KB

)(AS¬

α¬

)()( wSwP ∨¬

, , , ,

}/{ wy

)()( wRwS ∨

}/{ wx


Resolution example


KB

)(AS¬

α¬

)()( wSwP ∨¬

, , , ,

}/{ wy

)()( wRwS ∨

}/{ wx

)(wS

}/{ wz

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Resolution example


KB

)(AS¬

α¬

)()( wSwP ∨¬

, , , ,

}/{ wy

)()( wRwS ∨

}/{ wx

)(wS

}/{ wz

}/{ Aw

Contradictionα=|KB


Dealing with equality

• Resolution works for first-order logic without equalities• To incorporate equalities we need an additional inference rule• Demodulation rule

• Example:

• Paramodulation rule: more powerful• Resolution+paramodulation give a refutation-complete

proof theory for FOL

kii

k

tSUBSTtSUBSTSUBSTtt

φφφφσσφφφ

∨∨∨∨=∨∨

+− KK

K

11121

2121

)},,(/),(({,

failtUNIFY i ≠= ),( 1φσ

)()()),((

aPxxfafP =

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Sentences in Horn normal form• Horn normal form (HNF) in the propositional logic

– a special type of clause with at most one positive literal

• A clause with one literal, e.g. A, is also called a fact• A clause representing an implication (with a conjunction of

positive literals in antecedent and one positive literal in consequent), is also called a rule

• Generalized Modus ponens:

– is the complete inference rule for KBs in the Horn normal form. Not all KBs are convertible to HNF !!!

)()( DCABA ∨¬∨¬∧¬∨

))(()( DCAAB ⇒∧∧⇒Typically written as:


Horn normal form in FOL

First-order logic (FOL)– adds variables and quantifiers, works with terms

Generalized modus ponens rule:

Generalized modus ponens:• is complete for the KBs with sentences in Horn form;• Not all first-order logic sentences can be expressed in this

form

),(,',',' 2121

τστφφφφφφ

SUBSTnn ⇒∧∧ KK

),()',( s.t.on substituti a ii SUBSTSUBSTi φσφσσ =∀=

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Forward and backward chaining

Two inference procedures based on modus ponens for Horn KBs:• Forward chaining

Idea: Whenever the premises of a rule are satisfied, infer the conclusion. Continue with rules that became satisfied.Typical usage: If we want to infer all sentences entailed by the existing KB.

• Backward chaining (goal reduction)Idea: To prove the fact that appears in the conclusion of a rule prove the premises of the rule. Continue recursively.Typical usage: If we want to prove that the target (goal) sentence is entailed by the existing KB.

Both procedures are complete for KBs in Horn form !!!

α


Forward chaining example• Forward chaining

Idea: Whenever the premises of a rule are satisfied, infer the conclusion. Continue with rules that became satisfied

),(),(),( zxFasterzyFasteryxFaster ⇒∧

KB: R1:

R2:

R3:

Assume the KB with the following rules:),()()( yxFasterySailboatxSteamboat ⇒∧

),()()( zyFasterzRowBoatySailboat ⇒∧

)(TitanicSteamboat

)(MistralSailboat

)(PondArrowRowBoat

Theorem: ),( PondArrowTitanicFaster

F1:

F2:

F3:

?

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Forward chaining example

)(TitanicSteamboat)(MistralSailboat

KB: R1:R2:R3:

F1:F2:F3: )(PondArrowRowBoat


),()()( yxFasterySailboatxSteamboat ⇒∧),()()( zyFasterzRowBoatySailboat ⇒∧

?




KB: R1:R2:R3:

F1:F2:

),( MistralTitanicFaster

F3: )(PondArrowRowBoat

F4:



Rule R1 is satisfied:

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KB: R1:R2:R3:

F1:F2:



F4:



),( PondArrowMistralFasterRule R2 is satisfied:F5:





KB: R1:R2:R3:

F1:F2:



F4:



),( PondArrowMistralFasterRule R2 is satisfied:F5:Rule R3 is satisfied:

),( PondArrowTitanicFasterF6:


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Backward chaining example

• Backward chaining (goal reduction)Idea: To prove the fact that appears in the conclusion of a rule prove the antecedents (if part) of the rule & repeat recursively.

),()()( yxFasterySailboatxSteamboat ⇒∧



)(TitanicSteamboat

)(MistralSailboat

)(PondArrowRowBoat

KB:

Theorem: ),( PondArrowTitanicFaster

R1:

R2:

R3:

F1:

F2:

F3:



),( PondArrowTitanicFaster

)(TitanicSteamboat

R1

)(PondArrowSailboat

),()()( yxFasterySailboatxSteamboat ⇒∧

),( PondArrowTitanicFaster}/,/{ PondArrowyTitanicx

)(TitanicSteamboat

)(MistralSailboat

)(PondArrowRowBoat

F1:

F2:

F3:

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)(TitanicSteamboatR1


R2)(TitanicSailboat

)(PondArrowRowBoat)(PondArrowSailboat

),( PondArrowTitanicFaster}/,/{ PondArrowzTitanicy


)(TitanicSteamboat

)(MistralSailboat

)(PondArrowRowBoat

F1:

F2:

F3:



),( PondArrowyFaster)(TitanicSteamboat

R1


R2

)(PondArrowRowBoat),( yTitanicFaster

R3


R2

)(PondArrowRowBoat

)(PondArrowSailboat

)(MistralSailboat

)(MistralSailboat

)(TitanicSailboat

)(TitanicSteamboat

)(MistralSailboat

)(PondArrowRowBoat

F1:

F2:

F3:

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Backward chaining


R1


R2


R3


)(MistralSailboat

R2

)(PondArrowRowBoat

)(PondArrowSailboat

)(MistralSailboat

)(TitanicSailboat

y must be bound tothe same term


Backward chaining• The search tree: AND/OR tree• Special search algorithms exits (including heuristics): AO, AO*


R1

)(PondArrowSailboat


R2)(TitanicSailboat


R3

)(TitanicSteamboat R1

)(MistralSailboat

R2)(MistralSailboat

)(PondArrowRowBoat

first-order logic.milos/courses/cs2710-fall...– an interpretation gives a semantic to primitives....

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