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There Exist some Omega-Powers of Any Borel Rank

By Dominique Lecomte and Olivier Finkel

Abstract

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, 2007.International audienceOmega-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

Topics: 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]
Publisher: Springer
Year: 2007
OAI identifier: oai:HAL:ensl-00157204v1
Provided by: Hal-Diderot
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