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.Comment: To appear in the Proceedings of the 16th EACSL Annual Conference on
Computer Science and Logic, CSL 2007, Lausanne, Switzerland, September 11-15,
2007, Lecture Notes in Computer Science, (c) Springer, 200
