Optimizing Proof Search in Model Elimination

Many implementations of model elimination perform proof search by iteratively increasing a bound on the total size of the proof. We propose an optimized version of this search mode using a simple divide-and-conquer refinement. Optimized and unoptimized modes are compared, together with depth-bounded and best-first search, over the entire TPTP problem library. The optimized size-bounded mode seems to be the overall winner, but for each strategy there are problems on which it performs best. Some attempt is made to analyze why. We emphasize that our optimization, and other implementation techniques like caching, are rather general: they are not dependent on the details of model elimination, or even that the search is concerned with theorem proving. As such, we believe that this study is a useful complement to research on extending the model elimination calculus. A short version of this paper will appear in the Proceedings of the 13th International Conference on Automated Deduction (Rutgers University, 1996), published by Springer-Verlag in the Lecture Notes in Computer Science series. 1 Model elimination and PTTP For some time after its proposal by Loveland (1968), model elimination was pushed to the background by the intense flurry of activity in resolution theorem proving. It was given a new lease of life by Stickel’s work. A natural way of implementing model elimination calculi, Loveland’s MESON procedure in particular, is to adapt Prolog’s standard search strategy, viz. backward chaining on Horn clauses with unification and backtracking. We assume that the first order formula to be proved is negated and reduced to clausal form, so the task is to refute, i.e. prove falsity (?) from, an implicitly conjoined set of clauses, each one of the form: P1 _ : : : _ Pn Here each Pi is a literal, meaning either an atomic formula or the negation of one. Variables occurring in each clause are implicitly universally quantified. Now from each such clause, n+1 pseudo-Horn clause rules called ‘contrapositives’ are created.1 We will write ‘ ’ for a (syntactic) negating operation on literals; that is (:P ) is P , whereas P is :P for atomic P . First there are n rules of the form: P1 ^ : : : ^ Pi 1 ^ Pi+1 ^ : : : ^ Pn ) Pi and then there is one more of the form: P1 ^ : : : ^ Pn ) ? The idea is to use these rules in a Prolog-style backward proof of the goal ?. (We could further emphasize the Prolog connection by writing P :P1, ..., Pn instead of P1 ^ : : : ^ Pn ) P , and putting any variables in specific instances in upper case.) Stickel (1988) developed a Prolog Technology Theorem Prover (PTTP) based on just a few basic changes to a standard Prolog implementation: Perform sound unification. Most Prolog implementations omit an occurs check, allowing for example f(X) and X to be unified. This is, according to the logic programming folklore, necessary for efficiency reasons, or desirable in order to permit cyclic structures. At each stage, retain a list of the ancestor goals (i.e. those which have already been expanded on the path between ? and the current goal), and as well as the input rules, allow unification of the current goal with the negation of one of its ancestors. This gives an alternative way of solving a goal, instead of 1Recall that a Horn clause is a clause which contains at most one unnegated literal. Here we just single out each literal in turn to act as the head clause in a Prolog-style search, even if this literal and some or all of its antecedants in the clause are negative.