Exam 2 CS 451/551 Tuesday, April 11, 2000 This is an open-book, open-notes examination. There are four problems. Do not spend too much time on any problem. Read them all through first and solve them in the order that allows you to make the most progress. ------------------------------------------------------------------------------ Problem 1: A. Give the clauses for the following theorem: There was a party. Ann and Bill and Carl and Debbie were invited. Either Ann and Bill came or Carl and Debbie came (or both couples came). Joe claims that if Ann came to the party then Bill didn't come and if Debbie came then Carl didn't come. We want to prove that Joe is wrong. B. Give a proof of the theorem using the resolution inference system, with the set of support strategy. ------------------------------------------------------------------------------ Problem 2: A. Give the following theorem in first order logic: Everybody in our class has an older brother. Whenever a person is older than a second person, then the second one is younger than the first. Whenever a person is a brother of a second person, then the second one is a sibling of the first. Whenver a person is a sibling of a second person, and the first one is male, then the first person is a brother of the second. There is a male in our class. Therefore there is a person who has a younger brother. (Note: I did not say whether this person is in our class) Use the following predicates: C(x) - x is in our class O(x,y) - x is older than y Y(x,y) - x is younger than y B(x,y) - x is a brother of y S(x,y) - x is a sibling of y M(x) - x is male B. Give the clauses for the theorem. C. Give a proof of the theorem using the resolution inference system. ------------------------------------------------------------------------------ Problem 3: A. Give the step by step results of the unification algorithm for P(x,x,h(x)) and P(f(g(z)),f(y),z). B. Give the step by step results of the unification algorithm for Q(x,x,g(h(w)),w) and Q(f(g(z)),f(y),y,a). If you don't know how the algorithm works, you can get partial credit for giving the result of the algorithm. ------------------------------------------------------------------------------ Problem 4: Consider a resolution inference system with the set of support strategy. Recall how OTTER is implemented. At each point the "smallest" clause in the set of support is chosen and an inference is performed between this clause and every other possible clause. Suppose that OTTER had instead decided to choose each clause from the set of support in a depth first way (i.e., at each point try to do all inferences with one of the most recently generated clauses) This strategy is not complete. A. State what I mean when I say that the above strategy is not complete. B. Show that this strategy is not complete. ------------------------------------------------------------------------------