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Slothrop: Knuth-Bendix Completion with a Modern Termination Checker

  • Ian Wehrman
  • Aaron Stump
  • Edwin Westbrook
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4098)

Abstract

A Knuth-Bendix completion procedure is parametrized by a reduction ordering used to ensure termination of intermediate and resulting rewriting systems. While in principle any reduction ordering can be used, modern completion tools typically implement only Knuth-Bendix and path orderings. Consequently, the theories for which completion can possibly yield a decision procedure are limited to those that can be oriented with a single path order.

In this paper, we present a variant on the Knuth-Bendix completion procedure in which no ordering is assumed. Instead we rely on a modern termination checker to verify termination of rewriting systems. The new method is correct if it terminates; the resulting rewrite system is convergent and equivalent to the input theory. Completions are also not just ground-convergent, but fully convergent. We present an implementation of the new procedure, Slothrop, which automatically obtains such completions for theories that do not admit path orderings.

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References

  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    Bachmair, L.: Canonical Equational Proofs. In: Progress in Theoretical Computer Science. Birkhäuser (1991)Google Scholar
  3. 3.
    Bachmair, L., Dershowitz, N., Plaisted, D.A.: Completion Without Failure. In: Resolution of Equations in Algebraic Structures, Rewriting Techniques, vol. 2, pp. 1–30. Academic Press, London (1989)Google Scholar
  4. 4.
    Contejean, E., Marché, C., Urbain, X.: CiME3 (2004), available at: http://cime.lri.fr/
  5. 5.
    Filliâtre, J.-C.: Ocaml data structures, available at: http://www.lri.fr/~filliatr/software.en.html
  6. 6.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Automated Termination Proofs with AProVE. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 210–220. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Huet, G.: A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm. Journal of Computer and System Science 23(1), 11–21 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Knuth, D., Bendix, P.: Simple Word Problems in Universal Algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, Oxford (1970)Google Scholar
  9. 9.
    Löchner, B., Hillenbrand, T.: The Next Waldmeister Loop. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 486–500. Springer, Heidelberg (2002)Google Scholar
  10. 10.
    Sattler-Klein, A.: About Changing the Ordering During Knuth-Bendix Completion. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 176–186. Springer, Heidelberg (1994)Google Scholar
  11. 11.
    Stump, A., Löchner, B.: Knuth-Bendix Completion of Theories of Commuting Group Endomorphisms. In: Information Processing Letters (to appear, 2006)Google Scholar
  12. 12.
    Wehrman, I., Stump, A.: Mining Propositional Simplification Proofs for Small Validating Clauses. In: Armando, A., Cimatti, A. (eds.) 3rd International Workshop on Pragmatics of Decision Procedures in Automated Reasoning (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ian Wehrman
    • 1
  • Aaron Stump
    • 1
  • Edwin Westbrook
    • 1
  1. 1.Dept. of Computer Science and EngineeringWashington University in St. LouisSt. LouisUSA

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