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Some complete \(\omega \)-powers of a one-counter language, for any Borel class of finite rank

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Abstract

We prove that, for any natural number \(n\ge 1\), we can find a finite alphabet \(\Sigma \) and a finitary language L over \(\Sigma \) accepted by a one-counter automaton, such that the \(\omega \)-power

$$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$

is \({\varvec{\Pi }}^0_n\)-complete. We prove a similar result for the class \({\varvec{\Sigma }}^0_n\).

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Acknowledgements

We thank very much the anonymous referees for their very useful comments about a preliminary version of our article.

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Correspondence to Dominique Lecomte.

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Finkel, O., Lecomte, D. Some complete \(\omega \)-powers of a one-counter language, for any Borel class of finite rank. Arch. Math. Logic 60, 161–187 (2021). https://doi.org/10.1007/s00153-020-00737-4

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  • DOI: https://doi.org/10.1007/s00153-020-00737-4

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