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# Automates sur les ordres linéaires : Complémentation

Abstract : 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.
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https://tel.archives-ouvertes.fr/tel-00720658
Contributor : Chloé Rispal <>
Submitted on : Wednesday, July 25, 2012 - 12:19:00 PM
Last modification on : Wednesday, February 26, 2020 - 7:06:05 PM
Long-term archiving on: : Friday, October 26, 2012 - 2:40:49 AM

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• HAL Id : tel-00720658, version 1

### Citation

Chloé Rispal. Automates sur les ordres linéaires : Complémentation. Théorie et langage formel [cs.FL]. Université de Marne la Vallée, 2004. Français. ⟨tel-00720658⟩

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