There Exist some Omega-Powers of Any Borel Rank

TL;DR

This paper demonstrates that omega-powers of finitary languages can have any countable Borel rank, revealing their extensive topological complexity and filling a significant gap in understanding their classification.

Contribution

It proves that for every non-zero countable ordinal, there exist omega-powers that are complete at the corresponding Borel level, showing their vast topological diversity.

Findings

Existence of omega-powers with any non-zero countable Borel rank.

Construction of omega-powers that are complete at each Borel level.

Omega-powers exhibit a wide range of topological complexities.

Abstract

Omega-powers of finitary languages are languages of infinite words (omega-languages) in the form V^omega, where V is a finitary language over a finite alphabet X. They appear very naturally in the characterizaton of regular or context-free omega-languages. Since the set of infinite words over a finite alphabet X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). 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, that there exists a context free language L such that L^omega is analytic but not Borel, and that there exists a finitary language V such that V^omega is a Borel set of infinite rank. But it was still unknown which could be the possible infinite Borel ranks of omega-powers. We fill this gap here, proving the following very surprising result which shows that omega-powers exhibit a great topological complexity: for each non-null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers.