SCOTT: A Model-Guided Theorem Prover

SCOTT (Semantically Constrained Otter) is a resolution-based automatic theorem prover for first order logic. It is based on the high performance prover OTTER by W. McCune and also incorporates a model generator. This finds finite models which SCOTT is able to use in a variety of ways to direct its proof search. Clauses generated by the prover are in turn used as axioms of theories to be modelled. Thus prover and model generator inform each other dynamically. This paper describes the algor i thm and some sample results. SCOTT (Semantically Constrained Otter) is a resolution based automatic theorem prover for first order logic. So much is hardly revolutionary. What is new in SCOTT is the way in which it blends traditional theorem proving methods, best seen as purely syntactic, with techniques for semantic investigation more usually associated with constraint satisfaction problems. Thus it bridges two aspects of the science of reasoning. It was made by marrying an existing high-performance theorem prover to an existing model generator. Neither parent program was much modified in this process. The resulting combined system out-performs its parents on many problems, for some of which it is currently the most effective prover available. 1 T h e P a r e n t s 1.1 Syn tax : O T T E R The theorem prover OTTER, written by W. McCune and based on earlier work by E. Lusk, R. Overbeek and others, is a product of Argonne National Laboratory and is widely regarded as the most powerful program of its type for certain classes of problem ([McCune, 1990; Lusk and McCune, 1992]). Its basic method is forward chaining, applying a rule of inference R, seen here as a partial function on clauses, to generate new clauses as in Figure 1. The clauses are divided into two disjoint sets, the set of support and the usable list Initially the set of support is non-empty. Clearly the proof search may terminate with a successful proof, or the set of support may be emptied, showing that there is no proof in the Figure 1 : Basic O T T E R A l g o r i t h m chosen search space, or it may fail to terminate. First order logic being undecidable, this is to be expected. The simplest rule applied by OTTER is binary resolution, together with unification and factoring. More interesting and powerful variants include hyper-resolution, negative hyper-resolution and unit resulting resolution. For equational reasoning the available rules include the various forms of paramodulation and term rewriting (demodulation). A crucial part of the algorithm is the decision as to whether each deduced formula is "new". In general this means that it is not subsumed by any formula already kept. Much of the high performance of OTTER is due to its sophisticated techniques for reducing the time spent computing subsumption and related properties. These techniques are not the focus of the present paper. Also not shown in the simple version of the algor i thm in Figure 1 is the rewriting due to demodulation and the like which may take place before the subsumption test. Nor have back subsumption and back demodulation (whereby the generated clause is used to simplify the existing clause database) been made explicit, although in many applications they are important. 1.2 Semantics: F I N D E R Deduction is only one form of reasoning. Another is the generation of models of a given theory. A model of