Some Combinatorial Operators in Language Theory

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.Comment: 21 page

Multilinear representations of Free PROs

We describe a structure of PRO on hypermatrices. This structure allows us to define multilinear representations of PROs and in particular of free Pros. As an example of applications, we investigate the relations of the representations of Pros with the theory of automata.Comment: 39 page

Conception de modèles génériques pour les machines à états finis

Les travaux de ce mémoire s'inscrivent dans le cadre de la théorie des automates et s'articulent autour de la création d'un compilateur de machines à états finis, WFSC, permettant la création et la manipulation d'automates pondérés multibandes à classes de symboles avec identité et non-identité. Ce modèle de machine à états finis est obtenu par généralisation de l'étiquetage des transitions des automates à états finis. La première partie est consacrée à cette généralisation de l'étiquetage. Le but est d'augmenter le pouvoir d'expression et l'efficacité de leur représentation. Nous introduisons la notion de classe de symboles, puis donnons des définitions d'automates, transducteurs, automates multibandes étiquetés par classe de symboles et la notion de relation équi-contrainte comme extension des relations d'identité et de non-identité. La seconde partie est une contribution à l'algorithmique des machines à états finis. Nous étudions d'abord un certain nombre d'opérations pour les machines multibandes pondérées, comme la jointure et l'auto-intersection. Comme il existe une relation avec le problème de la correspondance de Post, nous décrivons une classe de relations rationnelles n-aires pour laquelle l'auto-intersection peut être calculée et donnons les algorithmes correspondants. Ensuite, nous présentons l'étude de la construction de l'automate des follows d'une expression rationnelle à l'aide de la structure ZPC. Une implémentation permettant de comparer cette construction par rapport à la construction classique, a été réalisée. Dans la dernière partie, nous décrivons l'implémentation des machines à états finis virtuelles dans XFST, qui applique le principe de l'évaluation paresseuse aux automates, puis la modélisation des automates pondérés multibandes à classes de symboles avec identité et non-identité dans WFSC. Une technique de programmation améliorant le comportement des classes polymorphes en C++, la bitwise virtuality, est décrite. Enfin, nous terminons par la description de quelques applications afin de démontrer la souplesse d'utilisation à la fois du compilateur et de la bibliothèque WFSC.The work presented in this thesis takes place in the scope of automata theory and connects with the creation of a finite state compiler, WFSC that allows the creation and the processing of symbol class multitape weighted automata with identity and non-identity. This model of finite state machine is obtained by extending the transition labeling of finite state automata. The first part is dedicated to that generalization. The aim is to increase the expressiveness, as well as the compactness of these machines. We introduce the notion of symbol classes and we give definitions for automata, transducers and multitape automata labeled with symbol classes. We introduce in addition the notion for equi-constrained relation that extend the identity and non-identity relations. The second part is a contribution to the algorithmic of finite state machines. At first time, we study a set of operations for weighted multitape automata like join, and the auto-intersection. Due to link with the Post correspondance problem, we describea class of n-ary rational relations for which the auto-intersection can be computed and we give the related algorithms. In second time, we present the construction of the follow automata of a rational expression through the ZPC structure. An implementation was made that allowed the comparison of our construction with respect to the classical one, showing its efficiency in practice. Finally, we describe the implementation of virtual finite state machine inside XFST, that apply the principle of lazy evaluation to finite state machine, and we describe how symbol class multitape weighted automata with identity are modeled in WFSC. A programming technique that improve the behavior of polymorphic classes in C+++, called bitwise virtuality, is described. We finish by describing some applications in order to demonstrate that both the compiler and the WFSC library are easy to use.ROUEN-BU Sciences (764512102) / SudocROUEN-BU Sciences Madrillet (765752101) / SudocTOURS-Polytech'Informat.Product. (372612209) / SudocSudocFranceF

Constrained Expressions and their Derivatives

This paper proposes an extension to classical regular expressions by the addition of two operators allowing the inclusion of boolean formulae from the zeroth order logic. These expressions are called constrained expressions. The associated language is defined thanks to the notion of interpretation and of realization. We show that the language associated when both interpretation and realization are fixed is stricly regular and can be not regular otherwise. Furthermore, we use an extension of Antimirov partial derivatives in order to solve the membership test in the general case. Finally, we show that once the interpretation is fixed, the membership test of a word in the language denoted by a constrained expression can be undecidable whereas it is always decidable when the interpretation is not fixed

Constrained Expressions and their Derivatives

This paper proposes an extension to classical regular expressions by the addition of two operators allowing the inclusion of boolean formulae from the zeroth order logic. These expressions are called constrained expressions. The associated language is defined thanks to the notion of interpretation and of realization. We show that the language associated when both interpretation and realization are fixed is stricly regular and can be not regular otherwise. Furthermore, we use an extension of Antimirov partial derivatives in order to solve the membership test in the general case. Finally, we show that once the interpretation is fixed, the membership test of a word in the language denoted by a constrained expression can be undecidable whereas it is always decidable when the interpretation is not fixed

Operads, quasiorders, and regular languages

International audienceWe generalize the construction of multi-tildes in the aim to provide double multi-tilde operators for regular languages. We show that the underlying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects. These operads are obtained from more primitive objects, namely precompositions, whose algebraic counterparts are investigated. One of these operads acts faithfully on languages in the sense that two different operators act in two different ways

Algorithms for the Join and Auto-Intersection of Multi-Tape Weighted Finite-State Machines.

International audienceA weighted finite-state machine with n tapes describes a rational relation on n strings. We recall some basic operations on n-ary rational relations, recast the important join operation in terms of "auto-intersection", and propose restricted algorithms for both operations. If two rational relations are joined on more than one tape, it can unfortunately lead to non-rational relations with undecidable properties. As a consequence, there cannot be a fully general algorithm, able to compile any rational join or auto-intersection. We define a class of triples 〈A,i,j〉 for which we are able to compile the auto-intersection of the machine A w.r.t. tapes i and j. We hope that this class is sufficient for many practical applications

Operads, quasiorders and regular languages

We generalize the construction of multitildes in the aim to provide multitilde operators for regular languages. We show that the underliying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects. These operads are obtained from more primitive objects, namely precompositions, whose algebraic counter-parts are investigated. One of these operads acts faithfully on languages in the sense that two different operators act in two different ways