AC Simplifications and Closure Redundancies in the Superposition Calculus

Abstract

Reasoning in the presence of associativity and commutativity (AC) is well known to be challenging due to prolific nature of these axioms. Specialised treatment of AC axioms is mainly supported by provers for unit equality which are based on Knuth-Bendix completion. The main ingredient for dealing with AC in these provers are ground joinability criteria adapted for AC. In this paper we extend AC joinability from the context of unit equalities and Knuth-Bendix completion to the superposition calculus and full first-order logic. Our approach is based on an extension of the Bachmair-Ganzinger model construction and a new redundancy criterion which covers ground joinability. A by-product of our approach is a new criterion for applicability of demodulation which we call encompassment demodulation. This criterion is useful in any superposition theorem prover, independently of AC theories, and we demonstrate that it enables demodulation in many more cases, compared to the standard criterion.