Generating small convergent systems can be extremely hard

Extended abstract
  • Klaus Madlener
  • Friedrich Otto
  • Andrea Sattler-Klein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)


A sequence \((R_{n,m} )_{n,m \in \mathbb{N}}\) of normalized string-rewriting systems on some finite alphabet is constructed such that, for all n, m ∃ ℕ,

  • Rn,m contains 44 rules, it is of size O(n+m), and it is compatible with a length-lexicographical ordering > on *, but

  • given the system R n,m and the ordering > as input, the Knuth-Bendix completion procedure will generate more than A(n, m) intermediate rules before a finite convergent system S n,m of size O(n+m) is obtained, where A denotes Ackermann's function.


Word Problem Critical Pair Finite Alphabet Finite Presentation Completion Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R.V. Book. Thue systems as rewriting systems. Journal Symbolic Computation 3, 39–68 (1987)Google Scholar
  2. 2.
    M. Jantzen. Confluent String Rewriting. EATCS Monographs on Theoretical Computer Science 14. Berlin: Springer 1988Google Scholar
  3. 3.
    D.E. Knuth, P. Bendix. Simple word problems in universal algebras. In: J. Leech (ed.): Computational Problems in Abstract Algebra. New York: Pergamon 1970, pp. 263–297Google Scholar
  4. 4.
    R.C. Lyndon, P.E. Schupp. Combinatorial Group Theory. Berlin: Springer 1977Google Scholar
  5. 5.
    R. McNaughton. Elementary Computability, Formal Languages, and Automata. Englewood Cliffs: Prentice-Hall 1982Google Scholar
  6. 6.
    C. O'Dunlaing. Undecidable questions related to Church-Rosser Thue systems. Theoretical Computer Science 23, 339–345 (1983)CrossRefGoogle Scholar
  7. 7.
    C. Sims. The Knuth-Bendix procedure for strings as a substitute for coset enumeration. Journal Symbolic Computation 12, 439–442 (1991)Google Scholar
  8. 8.
    J. Steinbach. Comparing on Strings: Iterated Syllable Ordering and Recursive Path Orderings. SEKI Report SR-89-15. Kaiserslautern: UniversitÄt 1989Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Klaus Madlener
    • 1
  • Friedrich Otto
    • 2
  • Andrea Sattler-Klein
    • 1
  1. 1.Fachbereich InformatikUniversitÄt KaiserslauternKaiserslauternWest Germany
  2. 2.Fachbereich Mathematik/InformatikGesamthochschule KasselKasselWest Germany

Personalised recommendations