58 research outputs found
Topological Complexity of omega-Powers : Extended Abstract
This is an extended abstract presenting new results on the topological
complexity of omega-powers (which are included in a paper "Classical and
effective descriptive complexities of omega-powers" available from
arXiv:0708.4176) and reflecting also some open questions which were discussed
during the Dagstuhl seminar on "Topological and Game-Theoretic Aspects of
Infinite Computations" 29.06.08 - 04.07.08
There Exist some Omega-Powers of Any Borel Rank
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.Comment: 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, 200
Topological Complexity of Locally Finite omega-Languages
to appear in Archive for Mathematical LogicInternational audienceLocally finite omega-languages were introduced by Ressayre in [Formal Languages defined by the Underlying Structure of their Words, Journal of Symbolic Logic, 53 (4), December 1988, p. 1009-1026]. These languages are defined by local sentences and extend omega-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite omega languages are analytic sets, the class LOC_omega of locally finite omega-languages meets all finite levels of the Borel hierarchy and there exist some locally finite omega-languages which are Borel sets of infinite rank or even analytic but non-Borel sets. This gives partial answers to questions of Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992] and of Duparc, Finkel, and Ressayre [Computer Science and the Fine Structure of Borel Sets, Theoretical Computer Science, Volume 257 (1-2), 2001, p.85-105]
An omega-power of a context-free language which is Borel above Delta^0_omega
We use erasers-like basic operations on words to construct a set that is both
Borel and above Delta^0_omega, built as a set V^\omega where V is a language of
finite words accepted by a pushdown automaton. In particular, this gives a
first example of an omega-power of a context free language which is a Borel set
of infinite rank.Comment: To appear in the Proceedings of the International Conference
Foundations of the Formal Sciences V : Infinite Games, November 26th to 29th,
2004, Bonn, Germany, Stefan Bold, Benedikt L\"owe, Thoralf R\"asch, Johan van
Benthem (eds.), College Publications at King's College (Studies in Logic),
200
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