Abstract
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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Beauquier, D., Nivat, M. (1985). About rational sets of factors of a bi-infinite word. In: Brauer, W. (eds) Automata, Languages and Programming. ICALP 1985. Lecture Notes in Computer Science, vol 194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015728
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DOI: https://doi.org/10.1007/BFb0015728
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