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A necessary condition for the rationality of the zeta function of a regular language. (English) Zbl 0675.68034
The zeta function of a formal language L is the series exp($\sum\sb{n\ge 1}a\sb nx\sp n/n)$, where $a\sb n$ is the number of words of length n in L. The author shows that if L is regular and if its zeta function is regular, then it has integer coefficients and each irreducible factor of its numerator and denominator divides the denominator of the generating function $\sum\sb{n\ge 0}a\sb nx\sp n$ of L (which is rational, L being regular). He shows, under the same hypothesis, that there are cyclic languages $L\sb 1$ and $L\sb 2$ such that the generating function G(L) of L is $G(L\sb 1)-G(L\sb 2)$ (a language is cyclic if (i) uv$\in L\Leftrightarrow vu\in L$ and (ii) $w\in L\Leftrightarrow w\sp n\in L$ (n$\ge 1))$. Moreover, it is decidable whether the zeta function of a regular language is rational.
Reviewer: Ch.Reutenauer

68Q70Algebraic theory of languages and automata
68Q45Formal languages and automata
11S40Zeta functions and $L$-functions of local number fields
Full Text: DOI
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