An ω-Power of a Finitary Language Which is a Borel Set of Infinite Rank

Article type: Research Article

Authors: Finkel, Olivier

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

Abstract: ω-powers of finitary languages are ω-languages in the form Vω, where V is a finitary language over a finite alphabet Sigma. Since the set Σω of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [13], by Simonnet [15], and by Staiger [18]. It has been proved in [14] that for each integer n≥1, there exist some ω-powers of context free languages which are Πn0-complete Borel sets, and in [5] that there exists a context free language L such that Lω is analytic but not Borel. But the question was still open whether there exists a finitary language V such that Vω is a Borel set of infinite rank. We answer this question in this paper, giving an example of a finitary language whose ω-power is Borel of infinite rank.

Keywords: infinite words, ω-languages, ω-powers, Cantor topology, topological complexity, Borel sets, infinite rank

Journal: Fundamenta Informaticae, vol. 62, no. 3-4, pp. 333-342, 2004