On The Unnecessity of Multiple Overlaps in Completion Theorem Proving

Abstract

Completion Theorem Proving is based on the observation that the task of proving a formula can be transformed into the task of solving a system of equations over a boolean polynomial ring. The latter can be accomplished by means of the completion of a set of rewrite rules obtained from the equational system. The central operation in this completion process is the generation of critical pairs comprising the computation of overlaps on products of atoms. This computation requires weak AC-unification, which is known to be NP-hard. Motivated by the THEOPOGLES system (Müller 1987), we show that Hsiang’s (1985) N-strategy can still be strengthened by precluding multiple overlaps. This results in a drastical reduction of the search space. Moreover, these systems manage with unification in a free theory, which can be performed in linear time. Since this special critical pair generation can be translated into a resolution step, provided the two rules correspond to clauses, our result also amounts to a generalization of Dietrich’s (1986) result on the translation from superposition steps into resolution steps.