1 These questions concern concern issues with substitution and Skolemization.

a. Given the premise Vx y P(x, y), it is not valid to conclude that q P(q, q). Give an example of a predicate P where the first is true but the second is false. b. Suppose that an inference engine is incorrectly written with the occurs check omitted, so that it allows a literal like P(x, F(x)) to be unified with P(q, q). (As mentioned, most standard implementations of Prolog actually do allow this.) Show that such an inference engine will allow the conclusion y P(q, q) to be inferred from the premise Vx y P(x, y).