On the separators on an infinite word generated by a morphism

  • Emmanuelle Garel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 900)


Let x be an infinite non periodic word on a finite alphabet A. For each position n, the separator of x at n is the smallest factor of x that starts at n and that does not appear before in x. Denote by Sx(n) the length of the separator of x at n and Sx the corresponding function. We consider the problem of computing Sx in the case where x is generated by iterating some morphism σ: A*→A*. We prove that if σ is q-uniform (q≥2) and x is circular then Sx is q-regular (in the sense of Allouche and Shallit [2], [23]), or, in other words that the corresponding formal power series that associates Sx(n) to the q-ary expression of n is rational (Salomaa, Soittola [22]).


Combinatoric on infinite words circularity q-regularity 


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  1. [1]
    J.P. Allouche. Automates Finis en Théorie des Nombres. Expo. Math., 5:239–266, 1987.Google Scholar
  2. [2]
    J.-P. Allouche and J. O. Shallit. The ring of k-regular sequences. Theorical Computer Science, 98:163–197, 1992.Google Scholar
  3. [3]
    J. Berstel. Fonctions rationnelles et addition. Actes de l'Ecole de Printemps de Théorie des Langages, LITP:177–183, 1981.Google Scholar
  4. [4]
    J. Berstel, C. Retenauer. Les séries rationnelles et leurs langages. Masson, 1984.Google Scholar
  5. [5]
    A. Blumer, J. Blumer, A. Ehrenfeucht, D. Haussler, M.T. Chen and J. Seiferas. The smallest automaton recognizing the subwords of a text. Theorical Computer Science, 40:31–55, 1985.Google Scholar
  6. [6]
    A. Blumer, J. Blumer, A. Ehrenfeucht, D. Haussler, M.T. Chen and J. Seiferas. atBuilding the Minimal DFA for the Set of all Subwords of a Word One-line in Linear Time. Proc. ICALP '84, Springer Verlag, LNCS-172:109–118, 1984.Google Scholar
  7. [7]
    G. Christol, T. Kamae, M. Mendés-France and G. Rauzy. Suites algébriques, automates et substitutions. Bull. Soc. Math. France, 108:401–419, 1980.Google Scholar
  8. [8]
    A. Cobham. Uniform tag sequences. Mathem. Syst. Theory, 6:164–192, 1972.Google Scholar
  9. [9]
    M. Crochemore. Optimal factor transducers, in A. Apostolico and Z. Galil. Combinatorics on words, Springer, Berlin, 31–43, 1984.Google Scholar
  10. [10]
    M. Crochemore. Transducers and repetitions. Theorical Computer Science, 45:63–86, 1984.Google Scholar
  11. [11]
    C. Davis, D. E. Knuth. Number representations and dragon curves. Journal of Recreational Mathematics, 3:2, 1982, April:66–81, 1970, 3:3, 1982, on july:133–149, 1970.Google Scholar
  12. [12]
    M. Dekking, M. Mendés-France, A. van der Poorten. FOLDS. Math. Intelligencer, 4:130–138, 4:190–195, 1982, Errata in Math. Intelligencer, 5:5, 1983.Google Scholar
  13. [13]
    Automata, Languages and Machines. Academic Press, vol A, 1974.Google Scholar
  14. [14]
    M. Gardner. An optimal parallel Mathematical Games. Sciences America, july:115–120, 1967.Google Scholar
  15. [15]
    M. Mendès-France. Courbes du dragon par pliage. The Mathematical Intelligencer, 4:815–866, 1983.Google Scholar
  16. [16]
    M. Mendès-France and J.O. Shallit. Wire Bending. Journal of Combinatorial Theory, Srie A, 50:1–23, 1989.Google Scholar
  17. [17]
    M. Mendès-France and G. Tenenbaum. Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro. Bulletin de la SMF, 2, 109:143–268, 1981.Google Scholar
  18. [18]
    F. Mignosi and P. Séébold. If a DOL Language is k-Power Free then it is Circular. Publications de l'université d'Amiens.Google Scholar
  19. [19]
    B. Mossé. Puissances de mots et reconnaissabilité des points fixes d'une substitution. Theorical Computer Science, 2, 99:327–334, 1992.Google Scholar
  20. [20]
    B. Mossé. Notions de reconnaissabilité pour les points fixes substitutions et complexit des suites automatiques. Publications du LMD, Luminy, Marseille.Google Scholar
  21. [21]
    M. Queffélec. Substitutions Dynamical Systems Spectral Analysis. Springer Verlag, LNM-1294, 1987.Google Scholar
  22. [22]
    A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer Verlag, LNM-1294, 1987. Texts and monographs in computer science.Google Scholar
  23. [23]
    J. Shallit. A Generalisation of automatic sequences. Theorical computer Science, Springer Verlag, LNCS-61:1–16, 1988.Google Scholar
  24. [24]
    P. Séébold. Morphismes itérés, Mot de Morse et Mot de Fibonacci. C. R. Acad. Sc. Paris, 295, 1982.Google Scholar
  25. [25]
    T. Tapsoba. Complexité des suites automatiques, Thesis, Université d'Aix-Marseille II, 1987. 130Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Emmanuelle Garel
    • 1
  1. 1.LITP 4Paris Cedex 05France

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