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

TL;DR

This paper constructs an example of a finitary language whose omega-power is a Borel set of infinite rank, advancing understanding of the topological complexity of omega-powers.

Contribution

It provides the first example of a finitary language with an omega-power that is a Borel set of infinite rank, answering an open question in the field.

Findings

Existence of a finitary language with omega-power of infinite Borel rank

Omega-power can be more topologically complex than previously demonstrated

Advances understanding of the topological properties of omega-powers

Abstract

Omega-powers of finitary languages are omega languages in the form V^omega, where V is a finitary language over a finite alphabet X. Since the set of infinite words over X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers naturally arises and has been raised by Niwinski, by Simonnet, and by Staiger. It has been recently proved that for each integer n > 0, there exist some omega-powers of context free languages which are Pi^0_n-complete Borel sets, and that there exists a context free language L such that L^omega is analytic but not Borel. But the question was still open whether there exists a finitary language V such that V^omega is a Borel set of infinite rank. We answer this question in this paper, giving an example of a finitary language whose omega-power is Borel of infinite rank.