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
is \({\varvec{\Pi }}^0_n\)-complete. We prove a similar result for the class \({\varvec{\Sigma }}^0_n\).
Similar content being viewed by others
References
Autebert, J.-M., Berstel, J., Boasson, L.: Context free languages and pushdown automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1. Springer (1996)
Cohen, R.S., Gold, A.Y.: Theory of \(\omega \)-languages, parts one and two. J. Comput. Syst. Sci. 15, 169–208 (1977)
Duparc, J., Finkel, O.: An \(\omega \)-power of a context free language which is Borel above \({\Delta }_\omega ^0\). In: Proceedings of the International Conference Foundations of the Formal Sciences V: Infinite Games, November 26th to 29th, 2004, Bonn, Germany, volume 11 of College Publications at King’s College (Studies in Logic), pp. 109–122. London (2007)
Duparc, J.: Wadge hierarchy and Veblen hierarchy: part 1: Borel sets of finite rank. J. Symb. Log. 66(1), 56–86 (2001)
Finkel, O.: Topological properties of omega context free languages. Theor. Comput. Sci. 262(1–2), 669–697 (2001)
Finkel, O.: Borel hierarchy and omega context free languages. Theor. Comput. Sci. 290(3), 1385–1405 (2003)
Finkel, O.: An omega-power of a finitary language which is a Borel set of infinite rank. Fundam. Inf. 62(3–4), 333–342 (2004)
Finkel, O.: Borel ranks and Wadge degrees of omega context free languages. Math. Struct. Comput. Sci. 16(5), 813–840 (2006)
Finkel, O., Lecomte, D.: There exist some \(\omega \)-powers of any Borel rank. In: Proceedings of the 16th EACSL Annual International Conference on Computer Science and Logic, CSL 2007, Lausanne, Switzerland, September 11–15, 2007, volume 4646 of Lecture Notes in Computer Science, pp. 115–129. Springer (2007)
Finkel, O., Lecomte, D.: Classical and effective descriptive complexities of omega-powers. Ann. Pure Appl. Log. 160(2), 163–191 (2009). arXiv:0708.4176
Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Publishing Co., Reading (2001). (Addison-Wesley Series in Computer Science)
Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)
Lecomte, D.: Omega-powers and descriptive set theory. J. Symb. Log. 70(4), 1210–1232 (2005)
Moschovakis, Y.N.: Descriptive Set Theory. North-Holland Publishing Co., Amsterdam (1980)
Niwinski, D.: A problem on \(\omega \)-powers. In: 1990 Workshop on Logics and Recognizable Sets. University of Kiel (1990)
Perrin, D., Pin, J.-E.: Infinite Words, Automata, Semigroups, Logic and Games, volume 141 of Pure and Applied Mathematics. Elsevier, Amsterdam (2004)
Simonnet, P.: Automates et théorie descriptive. Ph.D. thesis, Université Paris VII (1992)
Staiger, L.: Hierarchies of recursive \(\omega \)-languages. Elektronische Informationsverarbeitung und Kybernetik 22(5–6), 219–241 (1986)
Staiger, L.: \(\omega \)-languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 339–387. Springer, Berlin (1997)
Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, volume B, Formal Models and Semantics, pp. 135–191. Elsevier, Amsterdam (1990)
Wadge, W.: Reducibility and determinateness in the Baire space. Ph.D. thesis, University of California, Berkeley (1983)
Acknowledgements
We thank very much the anonymous referees for their very useful comments about a preliminary version of our article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-020-00737-4