1. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering A Hole in Goal Trees Notes for: D.W. Loveland and M. Stickel. “A Hole in Goal Trees: Some Guidance from Resolution Theory.” IEEE Transactions on Computing, C-25, 4 (April 1976), pp.335-341. For CSCE 580 Sp03 Marco Valtorta

2. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Swimming Pool Problem Statement I have a swimming pool If I have a swimming pool and it does not rain, I will go swimming If I go swimming, I will get wet If it rains, I will get wet Prove that I will get wet a: I have a swimming pool b: I go swimming c: I get wet d: It rains

3. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Prolog Program with Negation a. b <- a, ~d. c <- b. c <- d. ?- c. There is no goal tree that establishes c! However, c logically follows from the four clauses above.

4. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering A Sound and Complete Proof Procedure Add contrapositives, in particular the contrapositive of the last rule : –~d <- ~c. Check whether the negation of a goal appears as ancestor of the goal. If this is the case, mark the goal as proved This extension is sound and complete: only atoms that follow from the clauses are marked as proved, and all atoms that follow from the clauses can be marked as proved

5. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering The Example, Revisited c b d ~d a established, because it is a fact ~c established, because c is an ancestor and c is proved, because c is an ancestor of ~c in the goal tree.

6. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Soundness, Completeness, and Variables The rationale for this new proof procedure, which extends the usual goal tree procedure, is as follows: “Either c is true or ~c is true. If ~c is true, then we can establish c, …, which is impossible. Thus, c is true. This is an argument by contradiction” [L&S,p.337]. This proof procedure is also complete The generalization to the non-propositional case, using unification (matching) is also sound and complete