An Effective Property of $\omega$-Rational Functions

TL;DR

This paper establishes an effective automata-theoretic Baire property for $\omega$-regular languages and applies it to demonstrate a new property of $\omega$-rational functions, enabling the construction of a dense subset where the function is continuous.

Contribution

It introduces a novel effective property of $\omega$-rational functions, showing how to find a dense subset where these functions are continuous, based on automata-theoretic methods.

Findings

Proves $\omega$-regular languages satisfy an effective Baire property.

Constructs a dense $f \Pi^0_2$-subset where the function is continuous.

Provides a method to obtain a deterministic B"uchi automaton for the subset.

Abstract

We prove that $\omega$-regular languages accepted by B\"uchi or Muller automata satisfy an effective automata-theoretic version of the Baire property. Then we use this result to obtain a new effective property of rational functions over infinite words which are realized by finite state B\"uchi transducers: for each such function $F: \Sigma^\omega \rightarrow \Gamma^\omega$, one can construct a deterministic B\"uchi automaton $\mathcal{A}$ accepting a dense ${\bf \Pi}^0_2$-subset of $\Sigma^\omega$ such that the restriction of $F$ to $L(\mathcal{A})$ is continuous.