On nontermination of Knuth-Bendix algorithm

  • Mikuláš Hermann
  • Igor Prívara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)


The well-known Knuth-Bendix completion algorithm which computes a confluent and finitely terminating term rewriting system from a given set of equations, can either terminate with success or abort or even nonterminate. Very little is known about the origin of nontermination of this algorithm. We study the structural properties of rewrite rules which cause nontermination. The notion of the crossed rules is introduced for these purposes. We look for sufficient conditions guaranteeing nontermination of algorithm in the presence of crossed rules. A special attention is devoted to a verifiable condition of such kind.


Critical Pair Verifiable Condition Meta Rule Simple Word Problem Subterm Property 
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  1. Av 84.
    Avenhaus,J.: On the termination of the Knuth-Bendix completion algorithm, 120/84, Universität Kaiserslautern, 1984Google Scholar
  2. De 85a.
    Dershowitz,N.: Termination of rewriting; Report R-85-1220, University of Illinois at Urbana-Champaign, presented at 1st Int. Conf. on Rewriting Techniques and Applications (Dijon, France, May 85)Google Scholar
  3. De 85b.
    Dershowitz,N.-Marcus,L.-Tarlecki,A.: Existence, uniqueness, and construction of rewrite systems; unpublished manuscript, 1985Google Scholar
  4. HP 85.
    Hermann,M.-Prívara,I.: On nontermination of Knuth-Bendix algorithm; Research report VUSEI-AR-OPS-3/85Google Scholar
  5. Hu 80a.
    Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems; JACM 27, 4 (1980), 797–821.Google Scholar
  6. Hu 80b.
    Huet, G.: A complete proof of correctness of the Knuth-Bendix completion algorithm; Rapport 25, INRIA, 1980, also JCSS 23 (1980), 11–21Google Scholar
  7. KB 70.
    Knuth,D.E.-Bendix,P.: Simple word problems in universal algebras; in Computational Problems in Abstract Algebra (Ed. J.Leech), Pergamon PressGoogle Scholar
  8. Ki 85.
    Kirchner,H.: Preuves par complétion dans les variétés d'algèbres; Thèse de doctorat d'Etat, Université de NancyGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Mikuláš Hermann
    • 1
  • Igor Prívara
    • 1
  1. 1.VUSEI-AR, CSBratislavaCzechoslovakia

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