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 Knuth-Bendix completion procedure $(\mathrm{K}\mathrm{B})$ with one of the given orderings. To gain efficiency, however, we develop new inference rules working on objects called nodes, which are data

Multi-Completion with Termination Tools

“…Completion with multiple reduction orders-referred to as multi-completion in the sequel-was proposed by Kondo and Kurihara [15]. For a set O = { 1 , .…”

“…The basic control of mkbTT is a multi-completion variant of a discount loop, very similar to the one originally proposed for completion with multiple reduction orders [15]. Pseudo-code describing the control loop is given in Fig.…”

“…Kurihara and Kondo [15] introduced the calculus of multi-completion which supports completion with multiple reduction orders at the same time. Basically, a deduction simulates parallel completion runs with the orders under consideration but gains efficiency by sharing inference steps among the parallel deductions.…”

“…In this paper we describe the underlying inference system, and present simulation and correctness results along with the proofs that were omitted in [21,22] due to reasons of space. After detecting a flaw in the fairness definition of [15], we newly contribute a corrected and explicit definition for MKBtt. We also present a novel completeness result for a variant of the MKBtt calculus that (to our knowledge) has not been achieved by previous completion approaches.…”

Multi-completion with Termination Tools (System Description)

“…In this paper we describe a new tool for performing KnuthBendix completion with automatic termination tools. It is based on two ingredients: (1) the inference system for completion with multiple reduction orderings introduced by Kurihara and Kondo (1999) and (2) the inference system for completion with external termination provers proposed by Wehrman, Stump and Westbrook (2006) and implemented in the Slothrop system. Our tool can be used with any termination tool that satisfies certain minimal requirements.…”