A General Framework for the Derivation of Regular Expressions

The aim of this paper is to design a theoretical framework that allows us to perform the computation of regular expression derivatives through a space of generic structures. Thanks to this formalism, the main properties of regular expression derivation, such as the finiteness of the set of derivatives, need only be stated and proved one time, at the top level. Moreover, it is shown how to construct an alternating automaton associated with the derivation of a regular expression in this general framework. Finally, Brzozowski's derivation and Antimirov's derivation turn out to be a particular case of this general scheme and it is shown how to construct a DFA, a NFA and an AFA for both of these derivations.Comment: 22 page

An optimal parallel algorithm to convert a regular expression into its Glushkov automaton

AbstractThe aim of this paper is to describe a CREW-PRAM optimal algorithm which converts a regular expression of size s into its Glushkov automaton in O(log s) time using O(s2log s) processors. This algorithm makes use of the star-normal form of an expression as defined by Brüggemann-Klein (1993) and is based on the sequential algorithm due to Ziadi et al. (1997) which computes an original representation of Glushkov automaton in O(s) time

Bottom Up Quotients and Residuals for Tree Languages

In this paper, we extend the notion of tree language quotients to bottom-up quotients. Instead of computing the residual of a tree language from top to bottom and producing a list of tree languages, we show how to compute a set of k-ary trees, where k is an arbitrary integer. We define the quotient formula for different combinations of tree languages: union, symbol products, compositions, iterated symbol products and iterated composition. These computations lead to the definition of the bottom-up quotient tree automaton, that turns out to be the minimal deterministic tree automaton associated with a regular tree language in the case of the 0-ary trees

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

AUTOMATE, a computing package for automata and finite semigroups

AUTOMATE is a package for symbolic computation on finite automata, extended rational expressions and finite semigroups. On the one hand, it enables one to compute the deterministic minimal automaton of the language represented by a rational expression or given by its table. On the other hand, given the transition table of a deterministic automaton, AUTOMATE computes the associated transition monoid. The regular D-classes structure, and many properties of the elements in the monoid are provided. The program AUTOMATE has been written in C and is quite portable. The user interface includes specialized editors for easy displaying of the computed results

Minimisation d'automates non-déterministes, recherche d'expressions dans un texte et comparaison de génomes

Cette thèse débute par la minimisation des automates non-déterministes. Je fournis la preuve d'une technique présentée sans démonstration par Sengoku ainsi que différentes heuristiques, basées sur le calcul de simulations d'états, combinant langages gauches et droits. Ce travail débouche sur une technique de réduction des automates de Büchi. Parallèlement, je m'intéresse à la maîtrise de la complexité en espace de la déterminisation en optimisant la déterminisation partielle. Les thèmes suivants sont plus applicatifs. Le premier concerne la recherche approchée d'expressions secondaires dans le génome au moyen de grammaires algébriques. Je présente une adaptation de l'algorithme de Valiant, puis un algorithme de type CYK pour la recherche approchée d'une hélice simple. Je termine par la recherche d'équipes de gènes communes entre différents génomes, dont un problème sous-jacent est la recherche de composantes connexes communes à plusieurs graphes. J'y présente notre nouvel algorithme traitant le cas de graphes d'intervalles.The initial topic of this thesis is automata minimization. I prove a technique for full minimization that was given unproved by Sengoku, together with heuristics based on state simulations, that combine left and right languages. This work provides a reduction technique for B\"uchi automata. On the other hand, I focus on managing the space complexity of determinisation by an optimized partial determinization.The following is more involved in practical applications. First, I focus on secondary expression search in genome, based on context-free grammars. I give an adaptation of Valiant's algorithm, and a CYK algorithm for single hairpin approximate search. Finally, I investigate gene-team search between several genomes. An underlying problem is the common connected set search between several graphs. I describe our new algorithm that is specific to interval graphs.ROUEN-BU Sciences Madrillet (765752101) / SudocROUEN-BU Sciences (764512102) / SudocSudocFranceF

Génération aléatoire et structure des automates à états finis

La génération aléatoire de structures combinatoires en plus de permettre de mieux connaître les comportements des objets que l'on génère, permet de tester les algorithmes basés sur ces structures. Dans le cas des automates déterministes nous donnons les algorithmes de génération qui construisent ces objets sur n'importe quel alphabet. Nous observons que quasiment tous les automates déterministes complets et accessibles sont minimaux. Dans le cas des automates non déterministes nous établissons un protocole de génération probabiliste qui maximise la taille des déterminisés des automates générés. Par ailleurs, nous formalisons la technique de déterminisation partielle. Nous établissons une structure de données, les recouvrements d'automates, qui permet de manipuler et de donner des propriétés des automates non déterministes. Nous en déduisons une technique qui réduit la complexité de l'algorithme de déterminisation exhaustif classique.Random generation of combinatoric structures allows one to test algorithms based on this structure, and to investigate the behavior of these structures. In the case of deterministic automata, we give the generation algorithms that allow us to build these objects on any alphabets. We show that almost all complete accessible deterministic automata are minimal. In the case of nondeterministic automata we establish a probabilistic generation protocol that maximise the deterministic automata associated with these nondeterministic automata. Finally we continue the progress in the use of determinization for the pattern-matching problem. We formalize the technique of the partial determinization. We establish a data structure: the deterministic cover. This structure allows one to manipulate and to give properties of non-deterministic automata. We deduce from this structure a technique that reduces the complexity of the classical brute force determinization algorithm.ROUEN-BU Sciences Madrillet (765752101) / SudocROUEN-BU Sciences (764512102) / SudocSudocFranceF