Borel hierarchy and omega context free languages

  • Authors:

  • Olivier Finkel

  • Affiliations:

  • Equipe de Logique Mathématique, CNRS et Université Paris 7, U.F.R. de Mathématiques, 2 Place Jussieu 75251 Paris cedex 05, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

We give in this paper additional answers to questions of Lescow and Thomas (A decade of Concurrency, Lecture Notes in Computer Science, Vol. 803, Springer, Berlin, 1994, pp. 583-621), proving topological properties of omega context free languages (ω-CFL) which extend those of O. Finkel (Theoret. Comput. Sci. 262 (1-2) (2001) 669-697): there exist some ω-CFL which are non Borel sets and one cannot decide whether an ω-CFL is a Borel set. We give also an answer to a question of Niwinski (Problem on ω-Powers Posed in the Proceedings of the Workshop "Logics and Recognizable Sets, 1990") and of Simonnet (Automates et Thérie Descriptive, Ph.D. Thesis, Université Paris 7, 1992) about ω-powers of finitary languages, giving an example of a finitary context free language L such that Lω is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an ω-CFL is an arithmetical set. Finally we extend some results to context free sets of infinite trees.