There Exist some Omega-Powers of Any Borel Rank

Lecomte, Dominique ; Finkel, Olivier

HAL, ensl-00157204 / Harvested from HAL

Résumé

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.

Détail

Publié le : 2007-06-25
Classification:  Complete sets,  Infinite words,  Omega-languages,  Omega-powers,  Cantor topology,  Topological complexity,  Borel sets,  Borel ranks,  Complete sets.,  [INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO],  [INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC],  [INFO.INFO-CL]Computer Science [cs]/Computation and Language [cs.CL],  [MATH.MATH-LO]Mathematics [math]/Logic [math.LO]