Completion for multiple reduction orderings

Abstract

We present a completion procedure (called MKB) which works with multiple reduction orderings. Given equations and a set of reduction orderings, the procedure simulates a computation performed by the parallel processes each of which executes the standard Kuuth-Bendix completion procedure (KB) with one of the given orderings. To gain efficiency, however, we develop new inference rules working on objects called nodes, which are data structure consisting of a pair s: t of terms associated with the information to show which processes contain the rule s → t (or t → s) and which processes contain the equation s ↔ t. The idea is based on the observation that some of the inferences made in the processes are closely related, so we can design inference rules that simulate multiple KB inferences in several processes all in a single operation. Our experiments show that MKB is significantly more efficient than the naive simulation of parallel execution of KB procedures, when the number of reduction orderings is large enough.