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On the continuity setof an Omega rational function

Published online by Cambridge University Press:  18 January 2008

Olivier Carton,
Olivier Finkel and
Pierre Simonnet
Olivier Carton
Affiliation:
LIAFA, Université Paris 7 et CNRS, 2 Place Jussieu 75251 Paris Cedex 05, France; Olivier.Carton@liafa.jussieu.fr
Olivier Finkel
Affiliation:
Équipe Modèles de Calcul et Complexité,
Pierre Simonnet
Affiliation:
UMR 6134-Systèmes Physiques de l'Environnement, Faculté des Sciences, Université de Corse, Quartier Grossetti BP52 20250, Corte, France; simonnet@univ-corse.fr
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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 Σω for some alphabet Σ is the continuity set C(f) of an ω-rational synchronous function f defined on Σω .

Keywords

Infinitary rational relationsomega rational functionstopologypoints of continuitydecision problemsomega rational languagesomega context-free languages.

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