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Zeta functions of formal languages. (English) Zbl 0797.68092

The authors define the zeta function ζ(L) of a language L by ζ(L)=exp( n1 a n t n /n), where a n is the number of words of length n in L. They define the generalized zeta function Z(S) of a formal power series S in noncommuting variables by Z(S)=exp( n1 π(S n )/n), where π(S n ) is the homogeneous part of degree n of S viewed in a canonical way as a formal series in commuting variables. The generalized zeta function of a language L is Z(L ̲), where L ̲ is the characteristic series of L. By definition, a language L is cyclic if (i) uvL if and only if vuL and (ii) wL if and only if w n L (n1). The authors show that if L is a cyclic language then ζ(L)= n1 (1-t n ) -α n , where α n is the number of conjugation classes of primitive words of length n contained in L. If 𝒜 is a finite automaton over A, the trace tr(𝒜) of 𝒜 is the formal power series tr(𝒜)= wA * α w w, where α w is the number of couples (q,c) such that q is a state of 𝒜 and c is a path qq in 𝒜 labelled w. The authors prove that Z(tr(𝒜)) is rational. Using the theory of minimal ideals in finite semigroups, they are able to show that the characteristic series of a cyclic recognizable language is a linear combination over of traces of finite deterministic automata. Consequently, if L is cyclic and recognizable then ζ(L) and Z(L) are rational. As a corollary the authors obtain the rationality of the zeta function of a sofic system in symbolic dynamics. The authors discuss connections to Dwork’s theorem stating that the zeta function of an algebraic variety over a finite field is rational.

This remarkable paper opens up new and interesting research areas.


MSC:
68Q45Formal languages and automata
20M35Semigroups in automata theory, linguistics, etc.
05A15Exact enumeration problems, generating functions
37-99Dynamic systems and ergodic theory (MSC2000)
14G10Zeta-functions and related questions
37C25Fixed points, periodic points, fixed-point index theory
68Q70Algebraic theory of languages and automata