We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift.

A large number of properties of a vector addition system-for instance coverability, boundedness, or regularity-can be decided using its coverability graph, by looking for some characteristic pattern. We propose to unify the known exponential-space upper bounds on the complexity of such problems on vector addition systems, by seeing them as instances of the model-checking problem for a suitable extension of computation tree logic, which allows to check for the existence of these patterns. This provides new insights into what constitutes a "coverability-like" property.

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],

We study the "approximability" of unbounded temporal operators with time-bounded operators, as soon as some time bounds tend to ∞. More specif cally, for formulas in the fragments PLTL♦ and PLTL of the Parametric Linear Temporal Logic of Alur et al., we provide algorithms for computing the limit entropy as all parameters tend to ∞. As a consequence, we can decide the problem whether the limit entropy of a formula in one of the two fragments coincides with that of its time-unbounded transformation, obtained by replacing each occurrence of a time-bounded operator into its time-unbounded version. The algorithms proceed by translation of the two fragments of PLTL into two classes of discrete-time timed automata and analysis of their strongly-connected components.