Iterated substitutions and locally catenative systems: a decidability result in the binary case

  • Christian Choffrut
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 443)


Given a substitution h over a binary alphabet (a,b) and the infinite sequence a, h(a), h2(a),..., hn(a),..., we prove that for some m>0 the m-th term hm(a) of the sequence is an arbitrary product of the previous terms, if and only if it is already the case for m=2. This settles in a particular case the decidability of the local catenativity of the DOL-sequence, a longstanding open problem posed some fifteen years ago.

Key words

substitutions in the free monoids DOL-systems DOL-sequences linear recurrence equations locally catenative systems finite deciphering delay of codes equations in words 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

6. References

  1. [Be]
    J. Berstel, Some recent results on Square-Free Words, Lecture Notes on Computer Science no166, (1984), 14–25Google Scholar
  2. [BP]
    J. Berstel & D. Perrin, Theory of Codes, Academic Press, 1985.Google Scholar
  3. [BL]
    E. Barbin & M. Lerest, Sur la combinatoire des codes à deux mots, Theoret. Comput. Sci., 41, (1985), 61–80Google Scholar
  4. [CF]
    K. Culik II & I. Fris, The decidability of the equivalence problem for DOL-sequences, Inform. and Contr., 35, (1977), 20–39Google Scholar
  5. [ER1]
    A. Ehrenfeucht & G. Rozenberg, Elementary homomorphisms and a solution of the DOL equivalence problem, Theoret. Comput. Sci., 7,(1978), 169–183Google Scholar
  6. [ER2]
    A. Ehrenfeucht & G. Rozenberg, On the bound of two DOL-sequence equivalence problem, Theoret. Comput. Sci., 7, (1980), 169–183Google Scholar
  7. [HL]
    T. Harju & M. Linna, On the periodicity of morphisms on free monoids, RAIRO Theoretical Informatics and Applications, 20, (1986), 47–54Google Scholar
  8. [Ka1]
    J. Karhumäki, On the equivalence problem for binary DOL-systems, Inform. Contr., 50, (1981), 276–284Google Scholar
  9. [Ka2]
    J. Karhumäki, A property of three element codes, LNCS 166, Springer Verlag, (1984), 305–313Google Scholar
  10. [Ka3]
    J. Karhumäki, About the intersection of two free monoids generated by two elements, Semigroup Forum, (1984)Google Scholar
  11. [LR]
    A. Lindenmayer & G. Rozenberg, Developmental Systems with Locally Catenative Formulas, Acta Informatica, 2, (1973), 214–248Google Scholar
  12. [LS]
    A. Lentin & M.P. Schützenberger, A combinatorial problem in the theory of free monoids, Proc. University of North Carolina, (1967), 128–144Google Scholar
  13. [Ni]
    M. Nielsen, On the decidability of some equivalence problems for DOL-systems, Inform. Contr., 25, (1974), 166–193Google Scholar
  14. [Ru]
    K. Ruohonen, Remarks on locally catenative developmental sequences, Elektron. Informationsverarb. Kybernet., 14, (1978), 171–180Google Scholar
  15. [Sa]
    A. Salomaa, DOL-equivalence:the problem of iterated morphisms, EATCS Bull., 4, (1978), 5–12Google Scholar
  16. [Se]
    P. Séébold, About a family of binary morphisms whose stationary words are Sturmian, FCT'89, Springer LNCS 380, 384–399Google Scholar
  17. [Sh]
    J. Shallit, A generalization of automatic sequences, Theoret. Comput. Sci, 61, (1988), 1–16Google Scholar
  18. [Sp]
    J.C. Spehner, Quelques problèmes d'extension, de conjugaison et de présentation de sous-monoides d'un monoide libre, Thèse de Doctorat d'Etat, Paris, 1976Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Christian Choffrut
    • 1
  1. 1.Laboratorie d'Informatique de RouenUniversité de Rouen, Faculté des SciencesMont-Saint-Aignan CedexFrance

Personalised recommendations