On a zeta function associated with automata and codes

Abstract

The zeta function of a finite automaton A is exp{ ∞ n=1 a n z n n }, where a n is the number of bi-infinite paths in A labelled by a bi-infinite word of period n. It reflects the properties of A: aperiodicity, nil-simplicity, existence of a zero. The results are applied to codes.

The zeta functions of renewal systems

“…Thus S W (t) = exp (the first equality holds since W is pure and uniquely decipherable [12]), which completes the proof. 2…”

“…As for renewal systems, the zeta functions associated with codes are studied in [12]. The zeta function of the renewal system determined by a circular set coincides with the generating function of the circular set [2,17].…”