On the Topological Complexity of Infinitary Rational Relations

International audienceWe prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992]

On the Accepting Power of 2-Tape Büchi Automata

International audienceWe show that, from a topological point of view, 2-tape Büchi automata have the same accepting power than Turing machines equipped with a Büchi acceptance condition. In particular, we show that for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete infinitary rational relations accepted by 2-tape Büchi automata. This very surprising result gives answers to questions of W. Thomas [Automata and Quantifier Hierarchies, in: Formal Properties of Finite automata and Applications, Ramatuelle, 1988, LNCS 386, Springer, 1989, p.104-119] , of P. Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992], and of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In: "A Decade of Concurrency", LNCS 803, Springer, 1994, p. 583-621]

On the Continuity Set of an omega Rational Function

In this paper, we study the continuity of rational functions realized by B\"uchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function F has at least one point of continuity and that its continuity set C(F) cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational Pi^0_2-subset of X^omega for some alphabet X is the continuity set C(F) of an omega-rational synchronous function F defined on X^omega.Comment: Dedicated to Serge Grigorieff on the occasion of his 60th Birthda

Des codes pour engendrer des langages de mots infinis

This thesis deals with the languages of infinite words which are the ω-powers of a language of finite words. In particular, we focus on the open question : given a language L, does there exist an ω-code C such that C^ω = L^ω ? It is quite similar to the question deciding whether a submonoid of a free monoid is generated by a code.First, we study the set of relations satisfied by language L, i.e. the double factorizations of a word in L^∗ ∪ L^ω. We establish a necessary condition for that L^ω has a code or an ω-code generator. Next, we define the new class of languages where the set of relations is as simple as possible after codes : one-relation languages. For this class of languages, we characterize the languages L such that there exists a code or an ω-code C such that L^ω = C^ω, and we show that C is never a finite language. Finally, a characterization of codes concerning infinite words leads us to define reduced languages. We consider the properties of these languages as generators of languages of infinite words.Le sujet de cette thèse est l'étude des langages de mots infinis, en particulier les puissances infinies de langages de mots finis (puissance ω). Plus précisément, nous nous intéressons à la question ouverte suivante : étant donné un langage L, existe- t-il un ω-code C tel que C^ω = L^ω ? Cette question est l’analogue de celle pour la concaténation finie : un sous-monoïde d’un monoïde libre est-il engendré par un code ou non?Dans un premier temps, nous étudions l’ensemble des relateurs d’un langage L, c’est-`à-dire les couples de factorisations différentes d’un même mot de L^∗ ∪ ^Lω ; nous établissons une condition nécessaire pour que L^ω ait un code ou un ω-code générateur. Ensuite, nous définissons une nouvelle classe de langages : les langages à un relateur. Leurs ensemble de relateurs est le plus simple possible sans qu’ils soient des codes. Pour cette classe intéressante de langages, on caractérise les langages L tels qu’il existe un ω-code ou un code C tels que L^ω = C^ω. On montre que C ne peut pas être un langage fini. Enfin, une caractéisation des codes concernant les mots infinis nous amène à définir les langages réduits ; nous considérons les propriétés de ces langages en tant que générateurs de langages de mots infinis

Modal logics on rational Kripke structures

This dissertation is a contribution to the study of infinite graphs which can be presented in a finitary way. In particular, the class of rational graphs is studied. The vertices of a rational graph are labeled by a regular language in some finite alphabet and the set of edges of a rational graph is a rational relation on that language. While the first-order logics of these graphs are generally not decidable, the basic modal and tense logics are. A survey on the class of rational graphs is done, whereafter rational Kripke models are studied. These models have rational graphs as underlying frames and are equipped with rational valuations. A rational valuation assigns a regular language to each propositional variable. I investigate modal languages with decidable model checking on rational Kripke models. This leads me to consider regularity preserving relations to see if the class can be generalised even further. Then the concept of a graph being rationally presentable is examined - this is analogous to a graph being automatically presentable. Furthermore, some model theoretic properties of rational Kripke models are examined. In particular, bisimulation equivalences between rational Kripke models are studied. I study three subclasses of rational Kripke models. I give a summary of the results that have been obtained for these classes, look at examples (and non-examples in the case of automatic Kripke frames) and of particular interest is finding extensions of the basic tense logic with decidable model checking on these subclasses. An extension of rational Kripke models is considered next: omega-rational Kripke models. Some of their properties are examined, and again I am particularly interested in finding modal languages with decidable model checking on these classes. Finally I discuss some applications, for example bounded model checking on rational Kripke models, and mention possible directions for further research

Determinization of transducers over infinite words

International audienceWe study the determinization of transducers over infinite words. We consider transducers with all their states final. We give an effective characterization of sequential functions over infinite words. We also describe an algorithm to determinize transducers over infinite words