RAIRO - Theoretical Informatics and Applications

Table of Contents - January 2008 - Volume 42, Issue 01 (A nonstandard spirit among computer scientists:a tribute to Serge Grigorieff at the occasion of his 60th birthday)  

Research Article

On the continuity set of an Omega rational function

Carton, Oliviera1, Finkel, Oliviera2 and Simonnet, Pierrea3

a1 LIAFA, Université Paris 7 et CNRS, 2 Place Jussieu 75251 Paris Cedex 05, France; Olivier.Carton@liafa.jussieu.fr

a2 Équipe Modèles de Calcul et Complexité,

a3 UMR 6134-Systèmes Physiques de l'Environnement, Faculté des Sciences, Université de Corse, Quartier Grossetti BP52 20250, Corte, France; simonnet@univ-corse.fr

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 Σ ω for some alphabet Σ is the continuity set C(f) of an ω-rational synchronous function f defined on Σ ω .

(Online publication January 18 2008)

Key Words:

  • Infinitary rational relations;
  • omega rational functions;
  • topology;
  • points of continuity;
  • decision problems;
  • omega rational languages;
  • omega context-free languages.

Mathematics Subject Classification:

  • 68Q05;
  • 68Q45;
  • 03D05