Completion for Multiple Reduction Orderings

  • Authors:

  • Masahito Kurihara;Hisashi Kondo

  • Affiliations:

  • Department of Electrical Engineering, Hokkaido Institute of Technology, Sapporo 006-8585, Japan e-mail: kurihara@hit.ac.jp;Department of Systems Engineering, Ibaraki University, Hitachi 316-8511, Japan e-mail: kondo@lily.dse.ibaraki.ac.jp

  • Venue:
  • Journal of Automated Reasoning
  • Year:
  • 1999

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Abstract

We present a completion procedure (called MKB) that works for 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 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 structures 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 inferences made in different processes are often closely related, so we can design inference rules that simulate these inferences 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. We also present an extension of this technique to the unfailing completion for multiple reduction orderings, which is useful in various areas of automated reasoning, including equational theorem proving.