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Automata calculating the complexity of automatic sequences. (Automates calculant la complexité de suites automatiques.) (French) Zbl 0815.11015

The complexity of a sequence on a finite alphabet is the function \(n \to \rho (n)\), where \(\rho (n)\) is the number of factors (blocks) of length \(n\) of the sequence. In the case where the sequence is a fixed point of a uniform injective substitution and where the sequence takes only two values, the author shows that if the sequence is minimal, then \(n \to \rho (n+1) - \rho (n)\) is generated by an automaton which is explicitly constructed. Note that, as indicated by the author, B. Mossé has obtained more general results (Preprint, LMD).
The paper ends with an example, the Thue-Morse sequence, for which the result of the author implies the previously known result of complexity [see A. de Luca and S. Varricchio, Theor. Comput. Sci. 63, 333-348 (1989; Zbl 0671.10050) and S. Brlek, Discrete Appl. Math. 24, 83-96 (1989; Zbl 0683.20045)].

MSC:

11B85 Automata sequences
68Q45 Formal languages and automata
68R15 Combinatorics on words
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References:

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