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Course: CS 383, Fall 2009
School: UMass (Amherst)
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lecture Todays Lecture 13: Inference in FOL CMPSCI 383: Artificial Intelligence Instructor: Shlomo Zilberstein !! !! Knowledge representation in FOL Matching rules and facts using unification Inference procedures: forward-chaining, backward-chaining, and resolution Limitations of logical reasoning 2 !! !! 1 Shlomo Zilberstein University of Massachusetts Inference rules for FOL !! !! !! !! !! Handling...

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lecture Todays Lecture 13: Inference in FOL CMPSCI 383: Artificial Intelligence Instructor: Shlomo Zilberstein !! !! Knowledge representation in FOL Matching rules and facts using unification Inference procedures: forward-chaining, backward-chaining, and resolution Limitations of logical reasoning 2 !! !! 1 Shlomo Zilberstein University of Massachusetts Inference rules for FOL !! !! !! !! !! Handling quantifiers Conclusion q p, q p"q p#q p#r !! Modus Ponens: And-Elimination: And-Introduction: Or-Introduction: Resolution: Premises p, p ! q p"q p, q p p#q, q#r !! Want to instantiate sentences like: x < x + 1, owns(x,y) ! belongs(y,x) A substitution, Subst(\$, p), is a set of pairs that binds one pattern to another in p. !! !! Subst({f(x)/A}, f(g(f(x)))) Subst({x/John,y/Mary}, likes(x,y)) !! Given two atomic sentences p and q, \$ unifies p and q if Subst(\$, p) = Subst(\$, q) 4 Shlomo Zilberstein University of Massachusetts 3 Shlomo Zilberstein University of Massachusetts Unification example P Knows(John,x) Knows(John,x) Knows(John,x) F(x,y) Unification algorithm (defun sub-unify (pat-1 pat-2 theta) "Unify pat-1 with pat-2 given theta, the substitution so far" (cond ((equal pat-1 pat-2) theta) ((variable-p pat-1) (unify-var pat-1 pat-2 theta)) ((variable-p pat-2) (unify-var pat-2 pat-1 theta)) ((or (atom pat-1) (atom pat-2)) nil) ((eq (length pat-1) (length pat-2)) (let ((t* (sub-unify (first pat-1) (first pat-2) theta))) (if t* (sub-unify (rest pat-1) (rest pat-2) t*)))))) 5 Shlomo Zilberstein University of Massachusetts 6 Q Knows(y,Sarah) Knows(y,Mother(y)) Knows(x,Sarah) F(y,G(x)) ! {x/Sarah, y/John} {y/John, x/ Mother(John)} fail fail Shlomo Zilberstein University of Massachusetts Unification algorithm cont. (defun unify-var (var pattern theta) "Unifying a single variable" (let ((value (cdr (assoc var theta)))) (if value (sub-unify value pattern theta) (if (member var (variables (sublis theta pattern))) nil (acons var pattern theta))))) Universal elimination For any sentence P, variable x, and ground term g: %x P &' Subst({x/g},P) Examples: %x Likes(x,IceCream) &' Likes(Ben,IceCream) %x At(x,UMass) ! Smart(x) &' At(Bill,UMass) ! Smart(Bill) Shlomo Zilberstein University of Massachusetts 7 Shlomo Zilberstein University of Massachusetts 8 Existential elimination For any sentence P, variable x, and constant k that does not appear elsewhere in the KB: (x P &' Subst({x/k},P} Examples: (x Kill(x,Victim) &' Kill(Murderer,Victim) (x Smart(x) &' Smart(SmartPerson) Existential introduction For any sentence P, variable x that does not occur in P, and ground term g: P &' (x Subst({g/x},P) Examples: Likes(Jerry,IceCream) &' (x Likes(x,IceCream) Smart(Einstein) &' (x Smart(x) Shlomo Zilberstein University of Massachusetts 9 Shlomo Zilberstein University of Massachusetts 10 Finding proofs !! !! !! !! Horn sentences A Horn sentence has the form: P1 " P2 " ... " Pn ! Q where Pi and Q are positive atomic sentences Examples: Parent(g,p) " Parent(p,c) ! Grandparent(g,c) But? Sister(x,y) # Brother(x,y) ! Sibling(x,y) ) )Sibling(x,y) ! Sister(x,y) # Brother(x,y) 11 Shlomo Zilberstein University of Massachusetts 12 Finding a proof is a search process Operators/actions are the inference rules Search state is a set of sentences Using algorithms such as AO* for theorem proving Why is it an extremely hard search process? Shlomo Zilberstein University of Massachusetts Generalized Modus Ponens P1 " P2 " ... " Pn ! Q* P1, P2, ... Pn \$Q Can derive \$Q if \$ unifies Pi and Pi for all i Complete for Horn knowledge bases Forward chaining Shlomo Zilberstein University of Massachusetts 13 Shlomo Zilberstein University of Massachusetts 14 Properties of forward chaining !! Backward chaining !! !! !! Sound and complete for first-order definite clauses May not terminate in general if + is not entailed This is unavoidable: entailment with definite clauses is semi-decidable Matching itself can be expensive (matching conjunctive premises against known facts is NP-hard) 15 Shlomo Zilberstein University of Massachusetts 16 Shlomo Zilberstein University of Massachusetts Properties of backward chaining Resolution is sound and complete (p1 # ... # pm) (q1 # ... # qn) Subst(\$, p2 # ... # pm # q2 # ... # qn ) Provided that \$ unifies p1 and q1 Applies to KB in Conjunctive Normal Form e.g. 1. {P(x), Q(x,y)} GIVEN 2. {P(A), R(B,z)} GIVEN 3. {Q(A,y), R(B,z)} RESOLVE 1+2 Resolution is complete, theorem proving semi-decidable !! !! !! BUT, BC & FC combined are incomplete: e.g., A ! B, A ! C, C ! Z, B ! Z. We should be able to prove Z, but we cant. 17 Shlomo Zilberstein University of Massachusetts Shlomo Zilberstein University of Massachusetts 18 Conjunctive Normal Form Goal: KB contains only conjunction of disjunctions of literals (also called form). clausal Can convert any FOL KB into CNF (1) Remove implications Replace P ! Q by P#Q Replace P , Q by (P # Q) " (P # Q) Shlomo Zilberstein University of Massachusetts 19 Conjunctive Normal Form Cont. (2) )Move negation inwards %x P becomes (x P ) ) (x P becomes %x P P becomes P (P " Q) is replaced by P # Q (P # Q) is replaced by P " Q (3) Standardize variables each quantifier gets unique variables e.g. (xP(x)"(xQ(x) becomes (xP(x)"(yQ(y) 20 Shlomo Zilberstein University of Massachusetts Conjunctive Normal Form Cont. (4) (5) Move quantifiers to the left %xP # (yQ becomes %x(y P#Q Eliminate ( by Skolemization. (xP(x) becomes P(A) %x%y(zP(x,y,z) becomes %x%yP(x,y,F(x,y)) %x(yPred(x,y) becomes %xPred(x,Succ(x)) Drop universal quantifiers 21 Conjunctive Normal Form Cont. (7) Distribute And over Or (P " Q) # R becomes (P#R) " (Q#R) Remove operators (" #) (P # Q) " (P # R) becomes { {P,Q} {P,R} } (8) (6) Shlomo Zilberstein University of Massachusetts Shlomo Zilberstein University of Massachusetts 22 Resolution example Given: A. Twiddling someone guarantees they will twiddle you. B. Twiddling is transitive. C. Everyone twiddles somebody. Prove: D. Everyone twiddles themselves. Facts A. B. C. D. First-Order Logic %x%y T(x,y) ! T(y,x) %x%y%z T(x,y) " T(y,z) ! T(x,z) %x(y T(x,y) %x T(x,x) Shlomo Zilberstein University of Massachusetts 23 Shlomo Zilberstein University of Massachusetts 24 Sentences 1. 2. 3. 4. Clausal Form Derive a contradiction 1. 2. 3. 4. 5. 6. 7. 8. { T(x1,y1), T(y1,x1) } { T(x2,y2), T(y2,z2), T(x2,z2) } { T(x3,F(x3)) } { T(A,A) } { T(A,y2), T(y2,A) } 2+4 { T(F(A),A) } 3+5 { T(A,F(A)) } 1+6 {} 3+7 26 { T(x1,y1), T(y1,x1) } { T(x2,y2), T(y2,z2), T(x2,z2) } { T(x3,F(x3)) } { T(A,A) } (Negation of D) Shlomo Zilberstein University of Massachusetts 25 Shlomo Zilberstein University of Massachusetts Another resolution example !! !! !! !! !! Vincent has been murdered, and Arthur, Bertram, and Carleton are suspects. Arthur says he did not do it. He says that Bertram was the victim's friend but that Carleton hated the victim. Bertram says he was out of town the day of the murder, and besides he didn't even know the guy. Carleton says he is innocent and he saw Arthur and Bertram with the victim just before the murder. Assuming that everyone--except possibly for the murderer--is telling the truth, how ca we solve the crime? 27 Shlomo Zilberstein University of Massachusetts 28 Shlomo Zilberstein University of Massachusetts Converting the KB to CNF Shlomo Zilberstein University of Massachusetts 29 Shlomo Zilberstein University of Massachusetts 30 Resolving the crime Resolution strategies Deletion strategies: Clauses with certain properties are deleted before they are ever used. Examples: pure-literal elimination; tautology elimination; subsumption elimination (eliminate the more specific clause). Unit resolution...

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