There Exist some Omega-Powers of Any Borel Rank

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.
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Jacques Duparc and Thomas Henzinger. 16th EACSL Annual Conference on Computer Science and Logic, CSL 2007, September 11-15, 2007, Lausanne, Switzerland. Springer, pp.115-129, 2007, Lecture Notes in Computer Science
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  • ARXIV : 0706.3523

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Dominique Lecomte, Olivier Finkel. There Exist some Omega-Powers of Any Borel Rank. Jacques Duparc and Thomas Henzinger. 16th EACSL Annual Conference on Computer Science and Logic, CSL 2007, September 11-15, 2007, Lausanne, Switzerland. Springer, pp.115-129, 2007, Lecture Notes in Computer Science. 〈ensl-00157204〉

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