8 research outputs found
Classical and Effective Descriptive Complexities of omega-Powers
Final Version, published in A.P.A.L. This paper is an extended version of a conference paper which appeared in the Proceedings of the 16th EACSL Annual Conference on Computer Science and Logic, CSL 07. Part of the results in this paper have been also presented at the International Conference Computability in Europe, CiE 07, Siena, Italy, June 2007.International audienceWe prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232]
Classical and Effective Descriptive Complexities of omega-Powers
We prove that, for each non null countable ordinal alpha, there exist some
Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers,
extending previous works on the topological complexity of omega-powers. We
prove effective versions of these results. In particular, for each non null
recursive ordinal alpha, there exists a recursive finitary language A such that
A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do
this, we prove effective versions of a result by Kuratowski, describing a Borel
set as the range of a closed subset of the Baire space by a continuous
bijection. This leads us to prove closure properties for the classes
Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical
hierarchy in arbitrary recursively presented Polish spaces. We apply our
existence results to get better computations of the topological complexity of
some sets of dictionaries considered by the second author in [Omega-Powers and
Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p.
1210-1232].Comment: Final Version, published in A.P.A.L. This paper is an extended
version of a conference paper which appeared in the Proceedings of the 16th
EACSL Annual Conference on Computer Science and Logic, CSL 07. Part of the
results in this paper have been also presented at the International
Conference Computability in Europe, CiE 07, Siena, Italy, June 200
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
Highly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are -complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application
On Recognizable Languages of Infinite Pictures
An erratum is added at the end of the paper: The supremum of the set of Borel ranks of Büchi recognizable languages of infinite pictures is not the first non recursive ordinal but an ordinal which is strictly greater than the ordinal . This follows from a result proved by Kechris, Marker and Sami (JSL 1989).International audienceIn a recent paper, Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. The authors asked for comparing the tiling system acceptance with an acceptance of pictures row by row using an automaton model over ordinal words of length . We give in this paper a solution to this problem, showing that all languages of infinite pictures which are accepted row by row by Büchi or Choueka automata reading words of length are Büchi recognized by a finite tiling system, but the converse is not true. We give also the answer to two other questions which were raised by Altenbernd, Thomas and Wöhrle, showing that it is undecidable whether a Büchi recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable)