Positive deduction modulo regular theories

Abstract

We propose a new technique for dealing with an equational theory ɛ in the clausal framework. This consists of the definition of two inference rules called contextual superposition and extended superposition, and of an algorithm for computing the only needed applications of these last inference rules only by examining the axioms of ɛ. We prove the refutational completeness of this technique for a class of theories ɛ that include all the regular theories, i.e. any theory whose axioms preserve variables. This generalizes the results of Wertz [31] and Paul [17] who could not prove the refutational completeness of their superposition-based systems for any regular theory.