Inference in first-order logic
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1 CS 270 Foundations of AI Lecture 4 Inference in first-order logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square First-order logic FOL More epressive than propositional logic Advantages: Represents objects their properties relations and statements about them; Introduces variables that refer to an arbitrary objects and can be substituted by a specific object Introduces quantifiers allowing us to make statements over groups objects without the need to represent each of them separately
2 Order of quantifiers Order of quantifiers of the same type does not matter For all and y if is a parent of y then y is a child of y parent child y y parent child y Order of different quantifiers changes the meaning y loves Order of quantifiers Order of quantifiers of the same type does not matter For all and y if is a parent of y then y is a child of y parent child y y parent child y Order of different quantifiers changes the meaning y loves Everybody loves somebody y loves 2
3 Order of quantifiers Order of quantifiers of the same type does not matter For all and y if is a parent of y then y is a child of y parent child y y parent child y Order of different quantifiers changes the meaning y loves Everybody loves somebody y loves There is someone who is loved by everyone Connections between quantifiers Everyone likes ice cream? 3
4 Connections between quantifiers Everyone likes ice cream likes IceCream Connections between quantifiers Everyone likes ice cream likes IceCream Is it possible to convey the same meaning using an eistential quantifier? 4
5 Connections between quantifiers Everyone likes ice cream likes IceCream Is it possible to convey the same meaning using an eistential quantifier? There is no one who does not like ice cream likes IceCream A universal quantifier in the sentence can be epressed using an eistential quantifier!!! Connections between quantifiers Someone likes ice cream? 5
6 Connections between quantifiers Someone likes ice cream likes IceCream Is it possible to convey the same meaning using a universal quantifier? Connections between quantifiers Someone likes ice cream likes IceCream Is it possible to convey the same meaning using a universal quantifier? Not everyone does not like ice cream likes IceCream An eistential quantifier in the sentence can be epressed using a universal quantifier!!! 6
7 Representing knowledge in FOL Eample: Kinship domain Objects: people Mary Jane Properties: gender Male Female Relations: parenthood brotherhood marriage Parent Brother Spouse Functions: mother-of one for each person MotherOf 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 Male Female Parent and child relations are inverse y Parent Child y A grandparent is a parent of parent g c Grandparent g c p Parent g p Parent p c A sibling is another child of one s parents y Sibling p Parent p Parent p And so on. 7
8 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? KB? 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. 8
9 Logical inference problem in the Propositional logic Computational procedures that answer: KB? Three approaches: Truth-table approach Inference rules Conversion to the inverse SAT problem Resolution-refutation Inference in FOL: Truth table Is the Truth-table approach a viable approach for the FOL?? 9
10 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 Inference in FOL: Inference rules Is the Inference rule approach a viable approach for the FOL?? 0
11 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 by inference rules follows from the KB Caveat: we need to add rules for handling quantifiers Inference rules Inference rules from the propositional logic: Modus ponens A B A B Resolution A B B C A C and others: And-introduction And-elimination Orintroduction Negation elimination Additional inference rules are needed for sentences with quantifiers and variables Rules must involve variable substitutions
12 Sentences with variables First-order logic sentences can include variables. Variable is: Bound if it is in the scope of some quantifier P Free if it is not bound. Eamples: P Q y Likes Bound or free? y is free Sentences with variables First-order logic sentences can include variables. Variable is: Bound if it is in the scope of some quantifier P Free if it is not bound. P Q y is free Eamples: y Likes Bound Likes y Likes y Raymond Bound or free? 2
13 Sentences with variables First-order logic sentences can include variables. Variable is: Bound if it is in the scope of some quantifier P Free if it is not bound. P Q y is free Eamples: y Likes Bound Likes y Likes y Raymond is Bound first y is Free Sentences with variables First-order logic sentences can include variables. Sentence formula is: Closed if it has no free variables y P Q Open if it is not closed P Q y is free Ground if it does not have any variables Likes Jane 3
14 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 { 2 / t 2 / t } Application of the substitution to sentences SUBST { / Sam y / Pam} Likes Likes Sam Pam SUBST { / z y / fatherof } Likes? 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 { 2 / t 2 / t } Application of the substitution to sentences SUBST { / Sam y / Pam} Likes Likes Sam Pam SUBST { / z y / fatherof } Likes Likes z fatherof 4
15 Inference rules for quantifiers Universal elimination a - is a constant symbol a substitutes a variable with a constant symbol Likes IceCream Likes Ben IceCream Eistential elimination. a Substitutes a variable with a constant symbol that does not appear elsewhere in the KB Kill Victim Kill Murderer Victim Inference rules for quantifiers Universal instantiation introduction is not free in Introduces a universal variable which does not affect or its assumptions Sister Amy Jane Sister Amy Jane Eistential instantiation introduction a a is a ground term in is not free in Substitutes a ground term in the sentence with a variable and an eistential statement Likes Ben IceCream Likes IceCream 5
16 Unification Problem in inference: Universal elimination gives us many opportunities for substituting variables with ground terms a - is a constant symbol a Solution: avoid making blind substitutions of ground terms Make substitutions that help to advance inferences Use substitutions matching similar sentences in KB Make inferences on the variable level Do not substitute ground terms if not necessary Unification takes two similar sentences and computes the substitution that makes them look the same if it eists UNIFY p q s.t. SUBSTσ p SUBST q Unification. Eamples. Unification: UNIFY p q s.t. SUBSTσ p SUBST q Eamples: UNIFY Jane { / Jane} UNIFY y Ann? 6
17 7 Unification. Eamples. Unification: Eamples: } / { Jane Jane UNIFY } / / { y Ann Ann y UNIFY? y MotherOf y UNIFY s.t. q SUBST p SUBSTσ q p UNIFY Unification. Eamples. Unification: Eamples: } / { Jane Jane UNIFY } / / { y Ann Ann y UNIFY? Elizabeth UNIFY } / / { y MotherOf y MotherOf y UNIFY s.t. q SUBST p SUBSTσ q p UNIFY
18 Unification. Eamples. Unification: UNIFY p q s.t. SUBSTσ p SUBST q Eamples: UNIFY Jane { / Jane} UNIFY y Ann { / Ann y / } UNIFY y MotherOf { / MotherOf y / } UNIFY Elizabeth fail Generalized inference rules Use substitutions that let us make inferences!!!! Eample: Generalized Modus Ponens If there eists a substitution such that SUBST A SUBST A ' i i A A2 An B A ' A2 ' An ' SUBST B for all i=2 n 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 are tried 8
19 9 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 complete for the first-order logic and CNF w/o equalities if unsatisfiable the resolution will find the contradiction C B C A B A 2 2 n j j k i i n k SUBST fail UNIFY j i Eample:? y S Q Q P 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 complete for the first-order logic and CNF w/o equalities if unsatisfiable the resolution will find the contradiction C B C A B A 2 2 n j j k i i n k SUBST fail UNIFY j i Eample: y S P y S Q Q P
20 Inference with the resolution rule Proof by refutation: Prove that KB is unsatisfiable resolution is refutation-complete Main procedure steps:. Convert KB to CNF with ground terms and universal variables only 2. Apply repeatedly the resolution rule while keeping track and consistency of substitutions 3. Stop when empty set contradiction is derived or no more new resolvents conclusions follow Conversion to CNF. Eliminate implications equivalences p q p q 2. Move negations inside DeMorgan s Laws double negation p q p q p p p q p q p p p p 3. Standardize variables rename duplicate variables P Q P y Q 4. Move all quantifiers left no invalid capture possible P y Q y P Q 20
21 Conversion to CNF 5. Skolemization removal of eistential quantifiers through elimination If no universal quantifier occurs before the eistential quantifier replace the variable with a new constant symbol also called Skolem constant y P A Q P A Q B If a universal quantifier precedes the eistential quantifier replace the variable with a function of the universal variable y P Q P Q F F - a special function - called Skolem function Conversion to CNF 6. Drop universal quantifiers all variables are universally quantified P Q F P Q F 7. Convert to CNF using the distributive laws p q r p q p r The result is a CNF with variables constants functions 2
22 Resolution eample KB P w Q w Q S P R R z S z α SA Resolution eample KB P w Q w Q S P R R z S z α SA { y / w} P w S w 22
23 Resolution eample KB P w Q w Q S P R R z S z α SA { y / w} P w S w { / w} S w R w Resolution eample KB P w Q w Q S P R R z S z α SA { y / w} P w S w { / w} S w R w { z / w} Sw 23
24 Resolution eample KB P w Q w Q S P R R z S z α SA { y / w} P w S w { / w} S w R w { z / w} Sw { w/ A} Empty resolution Resolution eample KB P w Q w Q S P R R z S z α SA { y / w} P w S w { / w} S w R w { z / w} KB Sw { w/ A} Empty resolution Contradiction 24
25 Dealing with equality Resolution works for the first-order logic without equalities To incorporate equalities we need an additional inference rule Demodulation rule UNIFY z i t 2 k t t2 SUB SUBST t SUBST t fail where Eample: P f a f P a Paramodulation rule: more powerful Resolution+paramodulation give a refutation-complete proof theory for FOL 2 z i occurs in i 2 k 25
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