Geometrical regular languages and linear Diophantine equations: The strongly connected case

AbstractGiven an arbitrarily large alphabet Σ, we consider the family of regular languages over Σ for which the deterministic minimal automaton has a strongly connected state diagram. We present a new method for checking whether such a language is semi-geometrical or not and whether it is geometrical or not. This method makes use of the enumeration of the simple cycles of the state diagram. It is based on the construction of systems of linear Diophantine equations, where the coefficients are deduced from the set of simple cycles

Approximate Regular Expressions and Their Derivatives

International audienc

A generic method for the enumeration of various classes of directed polycubes

Combinatoric

Geometricity of Binary Regular Languages

International audienceOur aim is to present an efficient algorithm for checking whether a regular language is geometrical or not, based on specific properties of its minimal automaton. Geometrical languages have interesting theoretical properties and they provide an original model for off-line temporal validation of real-time softwares. As far as implementation is concerned, the regular case is of practical interest, which motivates the design of an efficient geometricity test addressing the family of regular languages. This study generalizes the algorithm designed by the authors for the case of prolongable binary regular languages

Langages géométriques et polycubes

Ce mémoire comporte deux parties. La première concerne l'étude des langages géométriques au moyen d'outils de la théorie des automates et de géométrie discrète. Un langage géométrique est composé de mots définis sur un alphabet de taille d, en utilisant les images de Parikh de l'ensemble des préfixes de ces mots. Ce qui définit une figure de dimension d. Dans la seconde partie, il est question de l'étude de polycubes de dimension 3. Il y est défini des extensions de certaines própriétés des polyominos en dimension 3. Cela permet de définir différentes classes de polycubes, les polycubes plateaux, s-dirigés et verticalement convexes s-dirigés. Une méthode d'énumération de polycubes dirigés, basée sur la décomposition par strates des polyominos de Temperley, est appliquée à ces classes de polycubes afin de donner leurs fonctions génératrices.This thesis falls into two parts. The first one is about the study of geometrical languages using formal languages and automata theory, as well as discrete geometry tools. A geometrical language is composed of words over an alphabet of size d, using the Parikh images of the set of prefixes of the words. Those images define a figure of dimension d. The second part refers to the study of 3-dimensional polycubes. We define 3-dimensional extensions of some properties of polyominoes. That allow us to define subclasses of polycubes : plateau polycubes, s-directed polycubes and vertically-convex s-directed polycubes. We define an enumeration method over directed polycubes, based on the strate decomposition of polyominoes defined by Temperley, and we use it in order to give the generating functions of the classes of polycubes defined above.ROUEN-BU Sciences Madrillet (765752101) / SudocSudocFranceF

A generic method for the enumeration of various classes of directed polycubes

CombinatoricsInternational audienceFollowing the track of polyominoes, in particular the column-by-column construction of Temperley and its interpretation in terms of functional equations due to Bousquet-Mélou, we introduce a generic method for the enumeration of classes of directed polycubes the strata of which satisfy some property P. This method is applied to the enumeration of two new families of polycubes, the s-directed polycubes and the vertically-convex s-directed polycubes, with respect to width and volume. The case of non-directed polycubes is also studied and it is shown that the generic method can be applied in this case too. Finally the general case of d-dimensional polycubes, with d≥4, is investigated, and the generic method is extended in order to handle the enumeration of classes of directed d-polycubes

Geometrical Regular Languages and Linear Diophantine Equations

International audienceWe present a new method for checking whether a regular language over an arbitrarily large alphabet is semi-geometrical or whether it is geometrical. This method makes use first of the partitioning of the state diagram of the minimal automaton of the language into strongly connected components and secondly of the enumeration of the simple cycles in each component. It is based on the construction of systems of linear Diophantine equations the coefficients of which are deduced from the the set of simple cycles. This paper addresses the case of a strongly connected graph

Enumeration of Specific Classes of Polycubes

International audienceThe aim of this paper is to gather several results concerning the enumeration of specific classes of polycubes. We first consider two classes of 3-dimensional vertically-convex directed polycubes: the plateau polycubes and the parallelogram polycubes. An expression of the generating function is provided for the former class, as well as an asymptotic result for the number of polycubes of each class with respect to volume and width. We also consider three classes of d-dimensional polycubes (d≥3) and we state asymptotic results for the number of polycubes of each class with respect to volume and width