Sur les mots sans carré définis par un morphisme

  • Jean Berstel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 71)


A word w is called repetitive if it contains two consecutive equal factors ; otherwise w is nonrepetitive. Thus the word abacacb is repetitive, and abcacbabcbac is nonrepetitive. There is no nonrepetitive word of length 4 over a two letter alphabet ; on the contrary, there exist infinite nonrepetitive words over a three letter alphabet. Most of the explicitly known infinite nonrepetitive words are constructed by iteration of a morphism. In this paper, we show that it is decidable whether an infinite word over a three letter alphabet obtained by iterating a morphism is nonrepetitive. We also investigate nonrepetitive morphisms, i.e. morphisms preserving nonrepetitive words, and we show that it is decidable whether a morphism (over an arbitrary finite alphabet) is nonrepetitive.


Letter Alphabet Infinite Word Primitive Word Nous Montrons Premier Article 
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.
    S.I. Adjan, Burnside groups of odd exponents and irreducible systems of group identities, in: Boone, Cannonito, Lyndon (eds), "Word Problems", North-Holland 1973, p 19–38.Google Scholar
  2. 2.
    S. Arson, Démonstration de l'existence de suites asymétriques infinies, Mat. Sb. 44 (1937), p 769–777.Google Scholar
  3. 3.
    J.M. Autebert, J. Beauquier, L. Boasson, M. Nivat, Quelques problèmes ouverts en théorie des langages algébriques, RAIRO, Informatique théorique, à paraître.Google Scholar
  4. 4.
    R. Dean, A sequence without repeats on x,x−1,y,y−1, Amer. Math. Monthly 72 (1965), p 383–385.Google Scholar
  5. 5.
    F. Dejean, Sur un théorème de Thue, J. Combinatorial Theory, Series A, 13 (1972), p 90–99.Google Scholar
  6. 6.
    A. Ehrenfeucht, K. Lee, G. Rozenberg, Subword complexities of various classes of deterministic developmental languages without interaction, Theor. Comput. Sci. 1 (1975), p 59–75.Google Scholar
  7. 7.
    W. Gottschalk, G. Hedlund, "Topological Dynamics", Amer. Math. Soc. Colloq. Publ. Vol. 36, 1955.Google Scholar
  8. 8.
    M. Harrison, "Introduction to Formal Language Theory", Addison-Wesley 1978.Google Scholar
  9. 9.
    S. Istrail, On irreductible languages and nonrational numbers, Bull. Soc. Math. Roumanie 21 (1977), p 301–308.Google Scholar
  10. 10.
    J. Leech, Note 2726: A problem on strings of beads, Math. Gazette 41 (1957), p 277–278.Google Scholar
  11. 11.
    M. Morse, G. Hedlund, Unending chess, symbolic dynamics and a problem in semigroups, Duke Math. J. 11 (1944), p 1–7.Google Scholar
  12. 12.
    P.A. Pleasants, Non-repetitive sequences, Proc. Cambridge Phil. Soc. 68 (1970), p 267–274.Google Scholar
  13. 13.
    C. Reutenauer, Sur les séries associées à certains systèmes de Lindenmayer, Theor. Comput. Sci., à paraître.Google Scholar
  14. 14.
    H.J. Shyr, A strongly primitive word of arbitrary length and its applications, Intern. J. Comput. Math., Section A 6 (1977), p 165–170.Google Scholar
  15. 15.
    A. Thue, Über unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat.-Nat. Kl., Christiania 1906, Nr. 7, p 1–22.Google Scholar
  16. 16.
    A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Vidensk. Skr. I. Mat.-Naturv. Kl., 1912, Nr. 1, p 1–67.Google Scholar
  17. 17.
    T. Zech, Wiederholungsfreie Folgen, Z. Angew. Math. Mech. 38 (1958), p 206–209.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • Jean Berstel
    • 1
  1. 1.Institut de ProgrammationUniversité Paris VI Laboratoire d'Informatique Théorique et Programmation, CNRSFrance

Personalised recommendations