Rewrite Methods for Clausal and Non-Clausal Theorem Proving

Effective theorem provers are essential for automatic verification and generation of programs. The conventional resolution strategies, albeit complete, are inefficient. On the other hand, special purpose methods, such as term rewriting systems for solving word problems, are relatively efficient but applicable to only limited classes of problems. In this paper, a simple canonical set of rewrite rules for Boolean algebra is presented. Based on this set of rules, the notion of term rewriting systems is generalized to provide complete proof strategies for first order predicate calculus. The methods are conceptually simple and can frequently utilize lemmas in proofs. Moreover, when the variables of the predicates involve some domain that has a canonical system, that system can be incorporated as rewrite rules, with the algebraic simplifications being done simultaneously with the merging of clauses. This feature is particularly useful in program verification, data type specification, and programming language design, where axioms can be expressed as equations (rewrite rules). Preliminary results from our implementation indicate that the methods are space-efficient with respect to the number of rules generated (as compared to the number of resolvents in resolution provers). 2. Introduct ion Given an equational theory E, a term rewriting system for E is a finite set of rewrite rules R.~--{li---*ri}~-_ 1 such that {l i=ri}~_ 1 and E are equivalent {i.e., s ~ t is true in {lr 1 if and only if s = t in E). A term t is reduced using rule I---.r if a subterm s of t, which is an instance of the left hand side l, is replaced by the corresponding instance of the right hand side r. A term s is reachable from t if t can be reduced to s after a finite number of reductions. A term is irreducible if no rule can be applied to it. We use t* to denote an irreducible form of t. We call a term rewriting system terminating if there is no infinite sequence of reductions from any term, and confluent if for any distinct terms t, r,