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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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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Equipe de Logique Mathématique, CNRS, Université Paris Diderot, Sorbonne Université, Campus des Grands Moulins, bâtiment Sophie-Germain, case 7012, 75205, Paris Cedex 13, France

    Olivier Finkel

  2. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Equipe d’Analyse Fonctionnelle, Sorbonne Université, Université Paris Diderot, CNRS, Campus Pierre et Marie Curie, case 247, 4, place Jussieu, 75252, Paris Cedex 5, France

    Dominique Lecomte

  3. Université de Picardie, I.U.T. de l’Oise, site de Creil, 13, allée de la faïencerie, 60107, Creil, France

    Dominique Lecomte

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  1. Olivier Finkel
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  2. Dominique Lecomte
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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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