RAIRO - Theoretical Informatics and Applications (RAIRO: ITA)

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RAIRO-Theor. Inf. Appl. 42, 183-196 (2008)
DOI: 10.1051/ita:2007050

On the continuity set of an Omega rational function

Olivier Carton¹, Olivier Finkel² and Pierre Simonnet³

¹ LIAFA, Université Paris 7 et CNRS, 2 Place Jussieu 75251 Paris Cedex 05, France; Olivier.Carton@liafa.jussieu.fr
² Équipe Modèles de Calcul et Complexité,
³ UMR 6134-Systèmes Physiques de l'Environnement, Faculté des Sciences, Université de Corse, Quartier Grossetti BP52 20250, Corte, France; simonnet@univ-corse.fr

(Published online: 18 January 2008)

Abstract
In this paper, we study the continuity of rational functions realized by Büchi 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 ${\bf\Pi}^0_2$ -subset of $\Sigma^\omega$ for some alphabet $\Sigma$ is the continuity set C(f) of an $\omega$ -rational synchronous function f defined on $\Sigma^\omega$ .

Mathematics Subject Classification. 68Q05, 68Q45, 03D05

Key words: Infinitary rational relations -- omega rational functions -- topology -- points of continuity -- decision problems -- omega rational languages -- omega context-free languages.

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