The word matching problem is undecidable for finite special string-rewriting systems that are confluent

  • Paliath Narendran
  • Friedrich Otto
Session 16: Rewriting
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)


We present a finite, special, and confluent string-rewriting system for which the word matching problem is undecidable. Since the word matching problem is the non-symmetric restriction of the word unification problem, this presents a non-trivial improvement of the recent result that for this type of string-rewriting systems, the word unification problem is undecidable (Otto 1995). In fact, we show that our undecidability result remains valid even when we only consider very restricted instances of the word matching problem.


matching unification equational theory string-rewriting systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BaSi94]
    F. Baader and J.S. Siekmann. Unification theory. In: D.M. Gabbay, C.J. Hogger, and J.A. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Oxford University Press, 1994.Google Scholar
  2. [BoOt93]
    R. Book and F. Otto. String-Rewriting Systems. Springer Verlag, New York, 1993.Google Scholar
  3. [Coc76]
    Y. Cochet. Church-Rosser congruences on free semigroups. Colloquia Mathematica Societatis János Bolyai 20 (1976) 51–60.Google Scholar
  4. [DeJo90]
    N. Dershowitz and J.P. Jouannaud. Rewrite systems. In: J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Vol. B: Formal Models and Semantics, Elsevier, Amsterdam, 1990, pages 243–320.Google Scholar
  5. [Jaf90]
    J. Jaffar. Minimal and complete word unification. Journal Association Computing Machinery 37 (1990) 47–85.Google Scholar
  6. [JoKi91]
    J.P. Jouannaud and C. Kirchner. Solving equations in abstract algebras: a rule-based survey of unification. In: J.L. Lassez and G. Plotkin (eds.), Computational Logic: Essays in Honor of Alan Robinson, MIT Press, 1991, pages 360–394.Google Scholar
  7. [KoPa]
    A. Kościelski and L. Pacholski. Makanin's group algorithm is not primitive recursive. Theoretical Computer Science, to appear.Google Scholar
  8. [Mak77]
    G.S. Makanin. The problem of solvability of equations in a free semigroup. Mat. Sbornik 103 (1977) 147–236.Google Scholar
  9. [Mak83]
    G.S. Makanin. Equations in a free group. Math. USSR Izvestija 21 (1983) 483–546.Google Scholar
  10. [Mak85]
    G.S. Makanin. Decidability of the universal and positive theories of a free group. Math. USSR Izvestija 25 (1985) 75–88.Google Scholar
  11. [MaAb94]
    G.S. Makanin and H. Abdulrab. On general solution of word equations. In: J. Karhumäki, H. Maurer, and G. Rozenberg (eds.), Results and Trends in Theoretical Computer Science, Lecture Notes Computer Science 812, Springer Verlag, Berlin, 1994, pages 251–263.Google Scholar
  12. [NaOt90]
    P. Narendran and F. Otto. Some results on equational unification. In: M.E. Stickel (ed.), Proceedings 10th CADE, Lecture Notes in Artificial Intelligence 449, Springer Verlag, Berlin, 1990, pages 276–291.Google Scholar
  13. [Ott95]
    F. Otto. Solvability of word equations modulo finite special and confluent string-rewriting systems is undecidable in general. Information Processing Letters 53 (1995) 237–242.CrossRefGoogle Scholar
  14. [PecS1]
    J.P. Pecuchet. Equations avec Constantes et Algorithme de Makanin. These 3e Cycle, Université de Rouen, France, Dec. 1981.Google Scholar
  15. [Sch90]
    K.U. Schulz. Makanin's algorithm for word equations-Two improvements and a generalization. In: K.U. Schulz (ed.), Word Equations and Related Topics, Proceedings, Lecture Notes Computer Science 572, Springer Verlag, Berlin, 1990, pages 85–150.Google Scholar
  16. [Sch93]
    K.U. Schulz. Word unification and transformation of generalized equations. Journal of Automated Reasoning 11 (1993) 149–184.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Paliath Narendran
    • 1
  • Friedrich Otto
    • 2
  1. 1.Institute of Programming and Logics, Department of Computer ScienceState University of New York at AlbanyAlbanyUSA
  2. 2.Fachbereich Mathematik/InformatikUniversität KasselKasselGermany

Personalised recommendations