An implementation of hyper-resolution

Abstract

In this paper the data structures and algorithms which were used to implement hyper-resolution are presented. The algorithms, which do not generate hyper-resolvents by creating sequences of P 1-resolvents, have been used to obtain proofs of THEOREM 1. Let G be a group such that x 3 = e for all x∈G. If h is defined as h(x, y) = xyx′y′ forx, y∈G, then for all x, y∈G, h(h(x, y), y) = e (the identity). THEOREM 2. Let R be a ring such that x 2 = x for all x∈R. Then R is commutative. THEOREM 3. Every subgroup of index 2 is normal. The data structures have been designed so that only a single copy of any literal or term is retained, no matter how often it occurs in the clauses kept. The main advantage of this approach is not the resulting savings in storage, but instead the fact that simultaneously matching a set of literals generates an entire set of hyper-resolvents. A method of extracting a set of “candidates for unification with a given literal” from the data structures is also presented. The result of using this method is a substantial reduction in the number of times a complete unification of two literals must be attempted. The initial results obtained from the program suggest that many resolution algorithms besides hyper-resolution could be enhanced by the use of similar data structures and algorithms.