The synchronized graphs trace the context-sensitive languages

International audienceMorvan and Stirling have proved that the context-sensitive languages are exactly the traces of graphs de ned by transducers with labelled nal states. We prove that this result is still true if we restrict to the traces of graphs de ned by synchronized transducers with labelled nal states. From their construction, we deduce that the context-sensitive languages are the languages of path labels leading from and to rational vertex sets of letter-to-letter rational graphs

Families of automata characterizing context-sensitive languages

International audienceIn the hierarchy of infinite graph families, rational graphs are defined by rational transducers with labelled final states. This paper proves that their traces are precisely context-sensitive languages and that this result remains true for synchronized rational graphs

Complementation of Rational sets on Scattered Linear Orderings of Finite Rank

International audienceIn a preceding paper [1], automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite, and even transfinite words studied by Buchi. Kleene's theorem has been generalized to these words. We show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation

Complementation of Rational Sets on Countable Scattered Linear Orderings

In a preceding paper (Bruyère and Carton, automata on linear orderings, MFCS'01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Büchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach

Complementation of Rational Sets on Scattered Linear Orderings

International audienceIn a preceding paper, automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-finite and even transfinite words studied by Buchi Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear ordering are closed under complementation using an algebraic approach

Automates sur les ordres linéaires : Complémentation

This thesis treats of rational sets of words indexed by linear orderings and particularly of the problem of the closure under complementation. In a seminal paper of 1956, Kleene started the theory of languages establishing that automata on finite words and rational expressions have the same expressive power. Since then, this result has been extended to many structures such as infinite words (Büchi, Muller), bi-infinite words (Beauquier, Nivat, Perrin), ordinal words (Büchi, Bedon), traces, trees... . More recently, Bruyère and Carton have introduced automata accepting words indexed by linear orderings and the corresponding rational expressions. Those linear structures include infinite words, ordinal words and their mirrors. Kleene's theorem has been generalized to words indexed by countable scattered linear orderings, that is orderings without any sub-ordering isomorphic to Q. For many structures, the class of rational sets forms a boolean algebra. This property is necessary to translate logic into automata. The closure under complementation was left as an open problem. In this thesis, we solve it in a positive way: we prove that the complement of a rational set of words indexed by scattered linear orderings is rational. The classical method to get an automaton accepting the complement of a rational set is trough determinization. We show that this method can not be applied in our case: An automaton is not necessary equivalent to a deterministic one. We have used other approaches. First, we generalize the proof of Büchi, based on congruence of words, to obtain the closure under complementation in the case of linear orderings of finite ranks. To get the whole result in the general case, we use the algebraic approach. We develop an algebraic structure extending the classical recognition by finite semigroups : semigroups are replaced by diamond-semigroups equipped with a generalized product. We prove that a set is rational iff it is recognized by a finite diamond-semigroup. We also prove that a canonical diamond-semigroup can be associated to each rational set : the syntactic diamond-semigroup. Our proof of the closure under complementation is effective. The theorem of Schützenberger establishes that a set of finite words is star-free if and only if its syntatic semigroup is finite and aperiodic. To finish, we partially extend this result to linear orderings of finite ranks.Cette thèse traite des ensembles rationnels de mots indexés par des ordres linéaires et en particulier du problème de la fermeture par complémentation. Dans un papier fondateur de 1956, Kleene initie la théorie des langages en montrant que les automates sur les mots finis et les expressions rationnelles ont le même pouvoir d'expression. Depuis, ce résultat a été étendu à de nombreuses structures telles que les mots infinis (Büchi, Muller), bi-infinis (Beauquier, Nivat, Perrin), les mots indexés par des ordinaux (Büchi, Bedon), les traces, les arbres... Plus récemment, Bruyère et Carton ont introduit des automates acceptant des mots indexés par des ordres linéaires et des expressions rationnelles correspondantes. Ces structures linéaires comprennent les mots infinis, les mots indexés par des ordinaux et leurs miroirs. Le théorème de Kleene a été généralisé aux mots indexés par les ordres linéaires dénombrables et dispersés, c'est-à-dire les ordres ne contenant pas de sous-ordre isomorphe à Q. Pour la plupart des structures, la classe des ensembles rationnels forme une algèbre de Boole. Cette propriété est nécessaire pour traduire une logique en automates. La fermeture par complémentation restait un problème ouvert. Dans cette thèse, on résout ce problème de façon positive: on montre que le complément d'un ensemble rationnel de mots indexés par des ordres linéaires dispersés est rationnel. La méthode classique pour obtenir un automate acceptant le complémentaire d'un ensemble rationnel se fait par déterminisation. Nous montrons que cette méthode ne peut-être appliquée dans notre cas: tout automate n'est pas nécessairement équivalent à un automate déterministe. Nous avons utilisé d'autres approches. Dans un premier temps, nous généralisons la preuve de Büchi, basée sur une congruence de mots, et obtenons ainsi la fermeture par complémentation dans le cas des ordres linéaires de rang fini. Pour obtenir le résultat dans le cas général, nous utilisons l'approche algébrique. Nous développons une structure algébrique qui étend la reconnaissance classique par semigroupes finis : les semigroupes sont remplacés par les diamant-semigroupes qui possèdent un produit généralisé. Nous prouvons qu'un ensemble est rationnel si et seulement s'il est reconnu par un diamant-semigroupe fini. Nous montrons aussi qu'un diamant-semigroupe canonique, appelé diamant-semigroupe syntaxique, peut être associé à chaque ensemble rationnel. Notre preuve de la complémentation est effective. Le théorème de Schützenberger établit qu'un ensemble de mots finis est sans étoile si et seulement si son semigroupe syntaxique est fini et apériodique. Pour finir, nous étendons partiellement ce résultat au cas des ordres de rang fini

Series-parallel languages on scattered and countable posets

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Schützenberger and Eilenberg theorems for words on linear orderings

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