Abstract
The operation V →V ω is a fundamental operation over finitary languages leading to ω-languages. Since the set Σ ω of infinite words over a finite alphabet Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers of finitary languages naturally arises and has been posed by Niwinski [Niw90], Simonnet [Sim92] and Staiger [Sta97a] . It has been recently proved that for each integer n ≥ 1, there exist some ω-powers of context free languages which are \({\bf \Pi}_n^0\)-complete Borel sets,[Fin01], that there exists a context free language L such that L ω is analytic but not Borel,[Fin03] , and that there exists a finitary language V such that V ω is a Borel set of infinite rank, [Fin04]. But it was still unknown which could be the possible infinite Borel ranks of ω-powers.
We fill this gap here, proving the following very surprising result which shows that ω-powers exhibit a great topological complexity: for each non-null countable ordinal ξ, there exist some -complete ω-powers, and some -complete ω-powers.
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Finkel, O., Lecomte, D. (2007). There Exist Some ω-Powers of Any Borel Rank. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_12
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DOI: https://doi.org/10.1007/978-3-540-74915-8_12
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