Design and Performance Evaluation of Knuth-Bendix Multi-Completion System Using Boolean Constrained Reduction Orders

  • Haruhiko Sato
  • Masahito Kurihara
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 33)


A multi-completion system MKB developed by Kurihara and Kondo accepts as input a set of reduction orders in addition to equations and efficiently simulates parallel processes each of which executes the standard completion procedure with one of the given orderings. In this paper, we describe a new system MKBOOL, which is a refinement of MKB in the sense that it restricts the reduction orders to those classes which can be represented by boolean constraints on some domains. Such classes include the recursive path orders (with status) for a finite signature. Our implementation represents a conjunction of the constraints by a binary decision diagram. The comprehensive experiments with the implementation on a set of well-known test problems show that in exchange for that restriction, MKBOOL runs more efficiently than MKB for most of the problems.


Term rewriting system Knuth-Bendix completion Multi-completion Recursive path ordering 



This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C), No. 19500020.


  1. 1.
    F. Baader and T. Nipkow. Term Rewriting and All That. Cambridge University Press, Cambridge, 1998.Google Scholar
  2. 2.
    L. Bachmair. Canonical Equational Proofs. Birkhäuser, Basel, 1991.zbMATHGoogle Scholar
  3. 3.
    R. E. Bryant. Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput., Vol. C-35, No. 8, pages 677–691, 1986.CrossRefGoogle Scholar
  4. 4.
    M. Codish, V. Lagoon and P. Stuckey. Solving partial order constraints for LPO termination. In Proc. 17th RTA, volume 4098 of LNCS, pages 4–18, 2006.Google Scholar
  5. 5.
    D. E. Knuth and P. B. Bendix. Simple word problems in universal algebras. in J. Leech (ed.), Computational Problems in Abstract Algebra, pages 263–297, Pergamon Press, New York, 1970.Google Scholar
  6. 6.
    M. Kurihara and H. Kondo. Completion for multiple reduction orderings. Journal of Automated Reasoning, Vol. 23, No.1, pages 25–42, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Kurihara and H. Kondo. BDD encoding for partial order constraints and its application to expert systems in software verification domains. In Proc. IEEE International Conference on Systems, Man and Cybernetics, pages 2062–2067, 2000.Google Scholar
  8. 8.
    J. Steinbach. Extensions and comparison of simplification orderings. in N. Dershowitz (ed.), Proc. 3rd RTA, volume 355 of LNCS, pages 434–448, 1989.MathSciNetGoogle Scholar
  9. 9.
    J. Steinbach and U. Kühler. Check your ordering - termination proofs and problems. Technical Report SR-90-25, Universität Kaiserslautern, 1990.Google Scholar
  10. 10.
    Terese. Term rewriting systems. Cambridge University Press, Cambridge 2003.Google Scholar
  11. 11.
    H. Zankl and A. Middeldorp. KBO as a satisaction problem. Technical Report, University of Innsbruck, 2006.Google Scholar

Copyright information

© Springer Science+Business Media B.V 2009

Authors and Affiliations

  • Haruhiko Sato
    • 1
  • Masahito Kurihara
    • 1
  1. 1.Graduate School of Information Science and TechnologyHokkaido UniversityJapan

Personalised recommendations