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2. Resolution for Predicate Logic CSE2303 Formal Methods I Lecture 23

3. Overview Program Clauses Resolution for Program Clauses Well Formed Formulae (WFF) Converting WFF to Clauses Resolution for Predicate Logic

4. Program Clauses Facts A. {{A}} Rules A:- B 1,…,B n {{A, ¬B 1,…, ¬B n }} Queries –Like conclusions. –So we negate them.

5. Database parent(bob, ann). parent(bob, pat). parent(pat, jim). {{parent(bob, ann)}} {{parent(bob, pat)}} {{parent(pat, jim)}}

6. ?- parent(bob, pat). Negate the query: {{¬parent(bob, pat)}} {parent(bob, pat)} {¬parent(bob, pat)}

7. ?- parent(X, ann). Negate the query: {{¬parent(X, ann)}} {¬parent(bob, ann)} {¬parent(X, ann)} X/bob {parent(bob, ann)}

8. Socrates mortal(X) :- man(X). man(socrates). ?- mortal(socrates). {{mortal(X), ¬man(X)}} {{man(socrates)}} {{¬mortal(socrates)}} All men are mortal Socrates is a man. Therefore Socrates is mortal.

9. ?- mortal(socrates). {mortal(socrates), ¬man(socrates)} X/socrates {mortal(X), ¬man(X)} {man(socrates)} {mortal(socrates)} {¬mortal(socrates)}

10. Natural Numbers If s(X) = X + 1 then all the natural numbers can be represented as 0, s(0), s(s(0)), s(s(s(0))),...

11. Addition For every X, Y, Z: X + 0 = X X + Y = Z  X + s(Y) = s(Z)

12. Addition add(X, 0, X). {{add(X, 0, X)}} add(X, s(Y), s(Z)) :- add(X, Y, Z). {{¬add(X, Y, Z), add(X, s(Y), s(Z))}} ?- add(s(0), s(0), X). {{¬add(s(0), s(0), X)}}

13. ? - add(s(0), s(0), X). {¬add(s(0), 0, Z)} {¬add(s(0), s(0), X)} {¬add(s(0), s(0), s(Z))} X /s(Z) {¬add(X, Y, Z), add(X, s(Y), s(Z))} {¬add(s(0), 0, Z), add(s(0), s(0), s(Z))} X/s(0), Y/0 {¬add(s(0), 0, s(0))} Z/s(0) {add(X, 0, X)} {add(s(0), 0, s(0) )} X/s(0)

14. Definitions A quantified sentence is called a well-formed formula (WFF) if every variable is quantified before being used. E.g.  X  Y (P(X,Y)   Z Q(X, Z)) (WFF)  X Q(Y) (not a WFF)  X (P(X, Y)   Y Q(X, Y)) (not a WFF)

15. Converting WFF into Clauses Eliminate  and  Move ¬ in as far as possible. Eliminate  (Skolemize) Eliminate  Put into clausal form

16. Moving ¬ ¬  X P(X) is logically equivalent to  X ¬P(X) ¬  X P(X) is logically equivalent to  X ¬P(X)

17. Love Example There is a girl who is loved by every boy.  X (girl(X)   Y(boy(Y)  loves(Y,X))) Therefore every boy loves some girl. Negating this we obtain. ¬  Y(boy(Y)   X (girl(X)  loves(Y,X)))

18.  X (girl(X)   Y(boy(Y)  loves(Y,X))) Eliminate   X (girl(X)   Y(¬boy(Y)  loves(Y,X))) Eliminate  girl(c)   Y(¬boy(Y)  loves(Y,c)) Eliminate  girl(c)  (¬boy(Y)  loves(Y,c)) Put into clausal form {{girl(c)}, {¬boy(Y), loves(Y,c)}}

19. ¬  Y(boy(Y)   X (girl(X)  loves(Y,X))) Eliminate  ¬  Y(¬boy(Y)   X (girl(X)  loves(Y, X))) Move ¬ in as far as possible.  Y(boy(Y)   X (¬girl(X)  ¬loves(Y, X))) Eliminate  boy(b)   X (¬girl(X)  ¬loves(b, X))) Eliminate  boy(b)  (¬girl(X)  ¬loves(b, X)) Put into clausal form {{boy(b)}, {¬girl(X), ¬loves(b, X)}}

20. Love Example There is a girl who is loved by every boy. {{girl(c)}, {¬boy(Y), loves(Y,c)}} Therefore every boy loves some girl. {{boy(b)}, {¬girl(X), ¬loves(b, X)}}

21. Proof {girl(c)} {¬loves(b,c)} {¬girl(X), ¬loves(b,X)} {¬girl(c), ¬loves(b,c)} X/c {¬boy(Y), loves(Y,c)} {¬boy(b), loves(b,c)} Y/b {boy(b)} {loves(b,c)}

22. Curiosity Example Jack owns a dog Every dog owner is an animal lover. No animal lover kills an animal. Either Jack or Curiosity killed the cat. The cat’s name is Tuna.

23. Jack owns a dog  X (dog(X)  owns(jack, X)) Every dog owner is an animal lover.  X ((  Y (dog(Y)  owns(X,Y)))  animalLover(X)) No animal lover kills an animal.  X, Y ((animalLover(X)  animal(Y))  ¬kills(X,Y)) Either Jack or Curiosity killed the cat. kills(jack, tuna)  kills(curiosity, tuna) The cat’s name is Tuna. cat(tuna) All cats are animals.  X cat(X)  animal(X) Formulae

24. Jack owns a dog {{dog(d)}, {owns(jack, d)}} Every dog owner is an animal lover. {{¬dog(Y), ¬owns(X, Y), animalLover(X)}} No animal lover kills an animal. {{¬animalLover(X), ¬animal(Y), ¬kills(X,Y)}} Either Jack or Curiosity killed the cat. {{kills(jack, tuna), kills(curiosity, tuna)}} The cat’s name is Tuna. {{cat(tuna)}} All cats are animals. {{¬cat(X), animal(X)}} Clausal Form

25. Questions Who killed tuna? {{¬kills(X, tuna)}} Did Curiosity kill the cat? {{¬kills(curiosity, X), ¬cat(X)}}

26. Revision Know the definition of Program Clauses. Be able to do resolution for Program Clauses. Be able to do resolution for Predicate Logic.