Regular languages and their generating functions: The inverse problem

Koutschan, Christoph

doi:10.1016/j.tcs.2007.10.031

Cited by 8 publications

(10 citation statements)

References 9 publications

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“…The converse by Soittola [108,23] of a theorem of Berstel [21] shows that it is possible to decide if a rational function is in fact an N-rational function. There is an effective version of this decidability result [16]; Koutschan [82] completed the details in order to get the first implementation of the algorithm. Giving an algorithm to decide if a function is N-algebraic, in a constructive way, would be nice.…”

Section: Discussionmentioning

confidence: 99%

Formulae and Asymptotics for Coefficients of Algebraic Functions

Banderier¹,

Drmota²

2014

Combinator. Probab. Comp.

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UUUWe dedicate this article to the memory of Philippe Flajolet, who was and will remain a guide and a wonderful source of inspiration for so many of us. UUU[ This article will appear in Combinatorics, Probability, and Computing, in the special volume dedicated to Philippe Flajolet. ] Cyril Banderier, CNRS/Univ. Paris 13, Villetaneuse (France). Cyril.Banderier at lipn.univ-paris13.fr, http://lipn.univ-paris13.fr/∼banderier Michael Drmota, TU Wien (Austria). drmota at dmg.tuwien.ac.at, http://dmg.tuwien.ac.at/drmota/ Date: March 22, 2013 (revised March 22, 2014. Key words and phrases. analytic combinatorics, generating function, algebraic function, singularity analysis, context-free grammars, critical exponent, non-strongly connected positive systems, Gaussian limit laws, N-algebraic function. 2 FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONSAbstract. We study the coefficients of algebraic functions n≥0 fnz n . First, we recall the too-little-known fact that these coefficients fn always admit a closed form. Then we study their asymptotics, known to be of the type fn ∼ CA n n α . When the function is a power series associated to a contextfree grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values for A. We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which states that α = −3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviors in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not N-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota-Lalley-Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for N-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).Résumé. Cet article a pour héros les coefficients des fonctions algébriques. Après avoir rappelé le fait trop peu connu que ces coefficients fn admettent toujours une forme close, nousétudions leur asymptotique fn ∼ CA n n α . Lorsque la fonction algébrique est la série génératrice d'une grammaire noncontextuelle, nous résolvons une vieille conjecture du folklore : les exposants critiques α ne peuvent pasêtre 1/3 ou −5/2 et sont en fait restreintsà un sousensemble des nombres dyadiques. Nous amorçons aussi l'étude de l'ensemble des valeurs possibles pour A. Nousétendons ce que Philippe Flajolet appelait le théorème de Drmota-Lalley-Woods (qui affirme que α = −3/2 dès lors qu'un "graphe de dépendance" associé au sys...

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Section: Discussionmentioning

confidence: 99%

Formulae and Asymptotics for Coefficients of Algebraic Functions

Banderier¹,

Drmota²

2014

Combinator. Probab. Comp.

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“…As previously mentioned, condition (18) ensures that all the labels of the succession rules equivalent to the given C-recurrence relation are positive, hence all the terms f n are positive. Thus it can be viewed as a sufficient condition to test the positivity of a given C-recurrence relation.…”

Section: Positivity Conditionmentioning

confidence: 98%

“…Soittola's Theorem has recently been proved in different ways in [8,22], using different approaches and some algorithms to pass from an N-rational series to a regular expression enumerated by such a series have been proposed [3,18]. However, none of these techniques provides a method to face C-finite recurrence relations which are not N-rational.…”

Section: Introductionmentioning

confidence: 99%

Recurrence relations, succession rules and the positivity problem

Bilotta

Pergola

Pinzani

et al. 2019

Journal of Computer and System Sciences

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In this paper we present a method to pass from a recurrence relation having constant coefficients (in short, a C-recurrence) to a finite succession rule defining the same number sequence. We recall that succession rules are a recently studied tool for the enumeration of combinatorial objects related to the ECO method. We also discuss the applicability of our method as a test for the positivity of a number sequence.

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“…This rational expression has been obtained with the package RLangGFun (Maple), created by Christoph Koutschan [8].…”

Section: Open Problemmentioning

confidence: 99%

“…We generalize the result of Mantel; indeed, we determine the necessary and sufficiently conditions which allow to decide if a formal series of the form (1 − ax + bx k ) −1 , where a ∈ N, b ∈ Z and k 2 ∈ N is N-rational or not. This result has been suggested by experimentations on computer with the package RLangGFun (Maple) of C. Koutschan [8]. In [2], Barcucci et al determine an algorithm for deciding if a rational series is N-rational.…”

Section: Introductionmentioning

confidence: 95%

-rationality of a certain class of formal series

Lavallée

2008

Information Processing Letters

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Regular languages and their generating functions: The inverse problem

Cited by 8 publications

References 9 publications

Formulae and Asymptotics for Coefficients of Algebraic Functions

Formulae and Asymptotics for Coefficients of Algebraic Functions

Recurrence relations, succession rules and the positivity problem

-rationality of a certain class of formal series

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