57 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
Borel Ranks and Wadge Degrees of Context Free Omega Languages
We show that, from a topological point of view, considering the Borel and the
Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power
than Turing machines equipped with a B\"uchi acceptance condition. In
particular, for every non null recursive ordinal alpha, there exist some
Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free
languages accepted by 1-counter B\"uchi automata, and the supremum of the set
of Borel ranks of context free omega languages is the ordinal gamma^1_2 which
is strictly greater than the first non recursive ordinal. This very surprising
result gives answers to questions of H. Lescow and W. Thomas [Logical
Specifications of Infinite Computations, In:"A Decade of Concurrency", LNCS
803, Springer, 1994, p. 583-621]
Wadge Degrees of -Languages of Petri Nets
We prove that -languages of (non-deterministic) Petri nets and
-languages of (non-deterministic) Turing machines have the same
topological complexity: the Borel and Wadge hierarchies of the class of
-languages of (non-deterministic) Petri nets are equal to the Borel and
Wadge hierarchies of the class of -languages of (non-deterministic)
Turing machines which also form the class of effective analytic sets. In
particular, for each non-null recursive ordinal there exist some -complete and some -complete -languages of Petri nets, and the supremum of
the set of Borel ranks of -languages of Petri nets is the ordinal
, which is strictly greater than the first non-recursive ordinal
. We also prove that there are some -complete, hence non-Borel, -languages of Petri nets, and
that it is consistent with ZFC that there exist some -languages of
Petri nets which are neither Borel nor -complete. This
answers the question of the topological complexity of -languages of
(non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326
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
Advances and applications of automata on words and trees : abstracts collection
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
08271 Abstracts Collection -- Topological and Game-Theoretic Aspects of Infinite Computations
From June 29, 2008, to July 4, 2008, the Dagstuhl Seminar 08271 ``Topological and Game-Theoretic Aspects of Infinite Computations\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, many participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
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