Thin homogeneous sets of factors

  • D. Beauquier
Session 4 Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 241)


This paper provides an algorithm to decide whether a set of words of length n is exactly the set of factors of length n of a unique bi-infinite word. In case of positive answer, this set of factors is said thin. We prove that a bi-infinite word u admits a thin set of factors of some length n iff u is periodic or ultimately periodic on the left and on the right but not with the same period. In other respects, as a tool for the proof, we give a standard form to the writing of a rational bi-infinite word which allows us to count easily its number of factors of length n, (i.e. : a words u such that {u} is the set recognized by a finite automation).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Beauquier D., Nivat M. (1985) About rational sets of factors of a bi-infinite word. Lecture Notes in Computer Science 194, 33–42 ICALP 85.Google Scholar
  2. [2]
    Ehrenfeucht A., Lee K.L., Rozenberg G. (1975) Subword complexities of various classes of deterministic developmental languages without interactions. Theoretical Computer Science 1, 59–75.Google Scholar
  3. [3]
    Ethan M. Coven, Headlund G.A. (1973) Sequences with Minimal Block Growth. Math System Theory 7, 138–153.Google Scholar
  4. [4]
    Lothaire M. (1983) Combinatorics on Words Encyclopedia of Mathematics and its applications, 17, 1–13.Google Scholar
  5. [5]
    Nivat M., Perrin D. (1982), Ensembles reconnaissables de mots bi-infinis Proc. 14th A.C.M. Symp. on Theory of Computing, 47–59.Google Scholar
  6. [6]
    Pansiot J.J., (1984), Bornes inférieures sur la complexité des facteurs des mots infinis engendrés par morphismes itérés, Lecture Notes in Computer Science, 166, STACS 84, p. 230–240.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • D. Beauquier
    • 1
  1. 1.L.I.T.P. — UER de Math.Université Paris 7Paris Cedex 05

Personalised recommendations