Descriptive Set Theory and $ω$-Powers of Finitary Languages

The $ω$-power of a finitary language L over a finite alphabet $Σ$ is the language of infinite words over $Σ$ defined by L $\infty$ := {w 0 w 1. .. $\in$ $Σ$ $ω$ | $\forall$i $\in$ $ω$ w i $\in$ L}. The $ω$-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{ü}chi automata or B{ü}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $ω$-powers.

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