Two Effective Properties of $\omega$-Rational Functions

Olivier Finkel (1, 2)

1 ELM - Equipe de Logique Mathématique
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

We prove two new effective properties of rational functions over infinite words which are realized by finite state B\"uchi transducers. Firstly, 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. Secondly, we give a new proof of the decidability of the first Baire class for synchronous $\omega$-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular $\omega$-languages.

Mots clés: Decision problems; regular languages of infinite words; infinitary rational relations; omega rational functions; topology; automatic Baire property; points of continuity; first Baire class; Wadge-Wagner class

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Olivier Finkel. Two Effective Properties of $\omega$-Rational Functions. International Journal of Foundations of Computer Science, 2021, 32 (7), pp.901-920. ⟨10.1142/S0129054121500283⟩. ⟨hal-01870467v4⟩