Sofic-Dyck Shifts

Béal, Marie–Pierre; Blockelet, Michel; Dima, Cătălin

doi:10.1007/978-3-662-44522-8_6

Cited by 10 publications

(38 citation statements)

References 32 publications

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“…In [12], Krieger and Matsumoto obtained an expression for the zeta function of a Markov-Dyck shift. The computation in [21] extends Krieger and Matsumoto's formula to sofic-Dyck shifts having a right-resolving presentation.…”

Section: Introductionmentioning

confidence: 58%

“…In Section III-B we describe how to obtained reduced presentations for sofic-Dyck which have good properties for the computation of the zeta function. In Section III-C, we recall the formula of the zeta function of a finite-type-Dyck shift obtained in [21]. It was proved in [21] for shifts having a right-resolving presentation and holds now for all finite-type-Dyck shifts.…”

Section: Introductionmentioning

confidence: 97%

“…In [21], we introduced the class of sofic-Dyck shifts which contains the above classes. We showed that they are characterized by a visibly pushdown language.…”

Section: Introductionmentioning

confidence: 99%

“…In [21] we performed a computation of the zeta function of a sofic-Dyck shift having a right-resolving presentation. The zeta function is a conjugacy invariant of shifts which counts the periodic sequences (see [23] for an overview of zeta functions).…”

Section: Introductionmentioning

confidence: 99%

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

Béal

Blockelet

Dima

2014

2014 Information Theory and Applications Workshop (ITA)

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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],

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Section: Introductionmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 97%

“…In [21], we introduced the class of sofic-Dyck shifts which contains the above classes. We showed that they are characterized by a visibly pushdown language.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Béal

Blockelet

Dima

2014

2014 Information Theory and Applications Workshop (ITA)

Self Cite

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show abstract

“…An equivalent semantics of Dyck automata is given in [7] with a graph semigroup associated to A. This graph semigroup is no more an inverse semigroup as for presentations associated to Markov-Dyck shifts [28].…”

Section: Sofic-dyck Shiftsmentioning

confidence: 99%