On the Continuity Set of an omega Rational Function

TL;DR

This paper investigates the continuity properties of omega rational functions, revealing undecidability of their continuity set and providing methods to compute or characterize these sets in specific cases.

Contribution

It proves the undecidability of the existence of at least one point of continuity for omega rational functions and characterizes the continuity set for synchronous functions.

Findings

Decidability of continuity for omega rational functions is limited.

Continuity set of synchronous rational functions is rational and computable.

Any rational Pi^0_2 subset can be realized as a continuity set of a synchronous omega-rational function.

Abstract

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.