845 research outputs found
Polishness of some topologies related to word or tree automata
We prove that the B\"uchi topology and the automatic topology are Polish. We
also show that this cannot be fully extended to the case of a space of infinite
labelled binary trees; in particular the B\"uchi and the Muller topologies are
not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the
proceedings of the 26th EACSL Annual Conference on Computer Science and
Logic, CSL 2017. The main addition with regard to the conference paper
consists in the study of the B\"uchi topology and of the Muller topology in
the case of a space of trees, which now forms Section
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 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
An Upper Bound on the Complexity of Recognizable Tree Languages
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class
for some natural number , where is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
into the class , and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
Decision Problems For Turing Machines
We answer two questions posed by Castro and Cucker, giving the exact
complexities of two decision problems about cardinalities of omega-languages of
Turing machines. Firstly, it is -complete to determine whether
the omega-language of a given Turing machine is countably infinite, where
is the class of 2-differences of -sets. Secondly,
it is -complete to determine whether the omega-language of a given
Turing machine is uncountable.Comment: To appear in Information Processing Letter
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
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