Mathematical systems theory

, Volume 18, Issue 1, pp 135–143 | Cite as

A note on thue systems with a single defining relation

  • Friedrich Otto
  • Celia Wrathall


A combinatorial characterization is given for those one-rule Thue systems of the form {(w1,w2)} with 0≦ |w2|≦|ov(w1)| that are Church-Rosser. Here ov(w1) denotes the longest proper self-overlap ofw1. Further, it is shown that a Thue system of this form is Church-Rosser if and only if there is an equivalent Thue system that is Church-Rosser.


Computational Mathematic Combinatorial Characterization Thue System 
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  1. 1.
    S. I. Adjan,Defining relations and algorithmic problems for groups and semigroups, Proceedings of the Steklov Institute of Mathematics 85 (1966), Amer. Math. Soc., 1967.Google Scholar
  2. 2.
    J. Avenhaus and K. Madlener, String matching and algorithmic problems in free groups,Revista Colombiana de Matematicas 14, 1–16 (1980).Google Scholar
  3. 3.
    J. Berstel, Congruences plus que parfaites et langages algébriques,Séminaire d'Informatique Théorique, Institut de Programmation, 1976–77, 123–147.Google Scholar
  4. 4.
    R. Book, Confluent and other types of Thue systems,J. Assoc. Comput. Mach. 29, 171–182 (1982).Google Scholar
  5. 5.
    R. Book, A note on special Thue systems with a single defining relation,Math. Systems Theory 16, 57–60 (1983).Google Scholar
  6. 6.
    R. Book, Homogeneous Thue systems and the Church-Rosser property,Discrete Mathematics 48, 137–145 (1984).Google Scholar
  7. 7.
    R. Book and C. O'Dúnlaing, Testing for the Church-Rosser property,Theoret. Comp. Sci. 16, 223–229 (1981).Google Scholar
  8. 8.
    Y. Cochet and M. Nivat, Une generalization des ensembles de Dyck,Israel J. Math. 9, 389–395 (1971).Google Scholar
  9. 9.
    G. Huet, Confluent reductions: Abstract properties and applications to term rewriting systems,J. Assoc. Comp. Mach. 27, 797–821 (1980).Google Scholar
  10. 10.
    D. Kapur, M. Krishnamoorthy, R. McNaughton, and P. Narendran, AnO(|T|3) algorithm for testing the Church-Rosser property of Thue systems,Theoret. Comp. Sci. 35, 109–114 (1985).Google Scholar
  11. 11.
    D. Knuth, J. Morris, and V. Pratt, Fast pattern matching in strings,SIAM J. Computing 6, 323–350 (1977).Google Scholar
  12. 12.
    G. Lallement,Semigroups and combinatorial applications, Wiley-Interscience, 1979.Google Scholar
  13. 13.
    G. Lallement, On monoids presented by a single relation,Journal of Algebra 32, 370–388 (1974).Google Scholar
  14. 14.
    M. Lothaire,Combinatorics on words, Addison-Wesley, 1983.Google Scholar
  15. 15.
    R. C. Lyndon and M. P. Schützenberger, The equationa M =b N c P in a free group,Mich. Math. J. 9, 289–298 (1962).Google Scholar
  16. 16.
    W. Magnus, A. Karrass, and D. Solitar,Combinatorial group theory, 2nd Revised Ed., Dover Publications, New York, 1976.Google Scholar
  17. 17.
    M. H. A. Newman, On theories with a combinatorial definition of equivalence,Annals Math. 43, 223–243 (1942).Google Scholar
  18. 18.
    M. Nivat (with M. Benois), Congruences parfaites et quasiparfaites,Séminaire Dubriel, 25e Année, 1971–72, 7-01-09.Google Scholar
  19. 19.
    M. O'Donnell,Computing in systems described by equations, Lecture Notes in Computer Science 58 (1977), Springer-Verlag.Google Scholar
  20. 20.
    C. O'Dúnlaing, Undecidable questions related to Church-Rosser Thue systems,Theoret. Comp. Sci. 23, 339–345 (1983).Google Scholar
  21. 21.
    C. O'Dúnlaing, Infinite regular Thue systems,Theoret. Comp. Sci. 25, 171–192 (1983).Google Scholar
  22. 22.
    B. Rosen, Tree manipulating systems and the Church-Rosser property,J. Assoc. Comp. Mach. 20, 160–187 (1973).Google Scholar

Copyright information

© Springer-Verlag New York Inc 1985

Authors and Affiliations

  • Friedrich Otto
    • 1
  • Celia Wrathall
    • 2
  1. 1.Fachbereich InformatikUniversität KaiserslauternKaiserslauternWest Germany
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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