Zeta functions of finite-type-Dyck shifts are N-algebraic
Marie-Pierre Béal, Michel Blockelet, Cătălin Dima · 2014
Constrained coding is a technique for converting unrestricted sequences of symbols into constrained sequences, i.e. sequences with a predefined set of properties. Regular constraints are described by finite-state automata and the set of bi-infinite constrained sequences are finite-type or sofic shifts. A larger class of constraints, described by sofic-Dyck automata, are the visibly pushdown constraints whose corresponding set of biinfinite sequences are the sofic-Dyck shifts. An algebraic formula for the zeta function, which counts the periodic sequences of these shifts, can be obtained for sofic-Dyck shifts having a right-resolving presentation. We extend the formula to all sofic-Dyck shifts. This proves that the zeta function of all sofic-Dyck shifts is a computable Z-algebraic series. We prove that the zeta function of a finite-type-Dyck shift is a computable N-algebraic series, i.e. is the generating series of some unambiguous context-free language. We conjecture that the result holds for all sofic-Dyck shifts.