Descriptive Set Theory and $\omega$-Powers of Finitary Languages

TL;DR

This paper explores the connections between Descriptive Set Theory and the structure of $oldsymbol{ extomega}$-powers of finitary languages, highlighting their relevance in automata theory and infinite word classification.

Contribution

It surveys recent results linking Descriptive Set Theory with $oldsymbol{ extomega}$-powers, advancing understanding of their role in automata and language classification.

Findings

$oldsymbol{ extomega}$-powers relate to complex descriptive set-theoretic classes

Recent results clarify the topological complexity of $oldsymbol{ extomega}$-powers

Connections inform automata-based characterizations of infinite languages

Abstract

The $\omega$-power of a finitary language L over a finite alphabet $\Sigma$ is the language of infinite words over $\Sigma$ defined by L $\infty$ := {w 0 w 1. .. $\in$ $\Sigma$ $\omega$ | $\forall$i $\in$ $\omega$ w i $\in$ L}. The $\omega$-powers appear very naturally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{\"u}chi automata or B{\"u}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $\omega$-powers.