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Dagstuhl Seminar Proceedings. 08271 - Topological and Game-Theoretic Aspects of Infinite Computations
Doi
Abstract
The operation of taking the omega-power Vomega of a language V is a fundamental operation over finitary languages leading to omega-languages. Since the set Xomega 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 Damian Niwinski (1990), Pierre Simonnet (1992), and Ludwig Staiger (1997). We investigate the topological complexity of omega-powers. We prove the following very surprising results which show that omega-powers exhibit a great opological complexity: for each non-null countable ordinal xi, there exist some Sigmax0​i-complete omega-powers, and some Pix0​i-complete omega-powers. On the other hand, the Wadge hierarchy is a great refinement of the Borel hierarchy, determined by Bill Wadge. We show that, for each ordinal xi greater than or equal to 3, there are uncountably many Wadge degrees of omega-powers of Borel rank xi+1. Using tools of effective descriptive set theory, we prove some effective versions of the above results
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