An Effective Property of $\omega$-Rational Functions

Abstract

We prove that $\omega$-regular languages accepted by Büchi 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 Büchi transducers: for each such function $F:{\Sigma }^{\omega }\to {\Gamma }^{\omega }$, one can construct a deterministic Büchi automaton $\mathcal{A}$ accepting a dense ${\mathbf{\Pi }}_2^0$-subset of ${\Sigma }^{\omega }$ such that the restriction of $F$ to $L(\mathcal{A})$ is continuous.

arXiv admin note: text overlap with arXiv:1107.5886