Brachman and Levesque, Chapter 4 exercise 1. Determine whether the following sentence is valid using resolution Summary: Brachman and Levesque, Chapter 4 exercise 1. Determine whether the following sentence is valid using resolution: xyz((P(y) Q(z)) (P(x) Q(x))) Answer To do this we need to check if from the negation of the sentence we can derive an empty clause (a contradiction). First transfer the negation into clausal form: ¬xyz((P(y) Q(z)) (P(x) Q(x))) ¬xyz(¬(¬P(y) Q(z)) (¬P(x) Q(x))) xyz¬(¬(¬P(y) Q(z)) (¬P(x) Q(x))) xyz(¬¬(¬P(y) Q(z)) ¬(¬P(x) Q(x))) xyz((¬P(y) Q(z)) (¬¬P(x) ¬Q(x))) xyz((¬P(y) Q(z)) P(x) ¬Q(x)) x((¬P(f(x)) Q(g(x))) P(x) ¬Q(x)) Clauses: C1 [¬P(f(x)), Q(g(x))] C2 [P(x)] C3 [¬Q(x)] Note that the definition of the resolution rule (p.58 of the textbook) pre- supposes that all clauses have distinct variables. (We can do this without loss of generality because variables are universally quantified, and xP(x) is equiv- Collections: Computer Technologies and Information Sciences