There Exist some Omega-Powers of Any Borel Rank

Dominique Lecomte (1) , Olivier Finkel (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 LIP - Laboratoire de l'Informatique du Parallélisme

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.

Mots clés: Complete sets; Infinite words; Omega-languages; Omega-powers; Cantor topology; Topological complexity; Borel sets; Borel ranks; Complete sets.

Domaines: Logique en informatique [cs.LO]; Complexité [cs.CC]; Informatique et langage [cs.CL]; Logique [math.LO]

Dominique Lecomte, Olivier Finkel. There Exist some Omega-Powers of Any Borel Rank. 16th EACSL Annual Conference on Computer Science and Logic, CSL 2007, September 11-15, 2007, Lausanne, Switzerland. pp.115-129. ⟨ensl-00157204⟩