Zeta functions of finite-type-Dyck shifts are N-algebraic

Abstract

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 rightresolving 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. I. INTRODUCTIONApplications of constrained coding are often confined to sequences drawn from regular languages. They are usually described by finite sets of forbidden blocks (finite-type constraints) or by finite-state automata (finite labelled graphs) and are then called sofic constraints since the set of these sequences is a symbolic dynamical system called a sofic shift [1].Nevertheless, there are classes of constrained sequences going beyond the sofic constraints (see for instance [2], [3]). A particular interesting one is the class of visibly pushdown sequences which corresponds to sofic-Dyck shifts and to visibly pushdown constraints. Visibly pushdown languages [4], [5] are a strict subclass of unambiguous context-free languages. They are rich enough to model many languages like XML languages. They form a natural and meaningful class in between the class of regular languages and the class of unambiguous context-free languages, extending the parenthesis languages [6], [7], the bracketed languages [8], and the balanced languages [9], [10]. The class of these languages is moreover tractable and robust. For instance the class of visibly pushdown languages over a same pushdown alphabet is stable by intersection and complementation.In [11], [12], [13], Inoue, Krieger, and Matsumoto introduced and studied special classes of shifts of sequences characterized by a context-free language of factors called Markov-Dyck shifts. They generalize the Dyck shifts whose blocks are finite factors of well parenthesized words. In [13],

Sofic-Dyck shifts

“…The process is an adaptation to Dyck automata of the determinization of visibly pushdown automata [1]. It is sketched in [8] and we detail it here.…”

“…It is proved in [8] that the zeta function of a finite-type-Dyck shifts is the generating series of an unambiguous context-free language, i.e. is an N-algebraic function.…”

On certain labelled directed graphs of symbolic dynamics

“…V ∈ B, σ ∈ B t(π(V,σ)) , π(V, σ)) = ǫ, V ′ ∈, σ ′ ∈ σ − π(V,σ) (V ′ )} as set of matching pairs. For information about invariants of topological conjugacy of finite-type-Dyck shifts see [BBD2].…”

“…This class contains a set of subshifts X that recode as finite-type-Dyck shifts X. Concerning invariants of topological conjugacy for finite-type-Dyck shifts see [BBD2].…”

Sofic-Dyck Shifts

“…The above formula shows that the zeta function of a sofic-Dyck shift is a Z-algebraic series. It is proved in [9] that the zeta function of a finite-type-Dyck shifts is the generating series of an unambiguous context-free language, i.e. is an N-algebraic function.…”