About rational sets of factors of a bi-infinite word

  • D. Beauquier
  • M. Nivat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 194)


In this paper, we consider the sets F(u) of finite factors of bi-infinite words u, upon a finite alphabet A. A natural question about this notion is : are there several bi-infinite words which have the same set of finite factors. We prove that, if F(u) is rational, then, there exists a unique bi-infinite word which has F(u) as set of finite factors iff F(u) has a non-exponential complexity (the complexity of F(u) is the function n — Card(F(u) ∩ An)) ; and if this condition is realized, in fact, F(u) has a sub-linear complexity (there exists a constant C such that card(F(u) ∩ An) ≤ n+C for large enough integers n).

Furthermore, the proof gives a characterization of bi-infinite rational words : u is rational iff F(u) is a rational set of non-exponential complexity.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • D. Beauquier
    • 1
  • M. Nivat
    • 1
  1. 1.L.I.T.P. — Université Paris VIIFrance

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