31 research outputs found

    Wadge Degrees of ω\omega-Languages of Petri Nets

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    We prove that ω\omega-languages of (non-deterministic) Petri nets and ω\omega-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ω\omega-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of ω\omega-languages of (non-deterministic) Turing machines which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal α<ω_1CK\alpha < \omega\_1^{{\rm CK}} there exist some Σ0_α{\bf \Sigma}^0\_\alpha-complete and some Π0_α{\bf \Pi}^0\_\alpha-complete ω\omega-languages of Petri nets, and the supremum of the set of Borel ranks of ω\omega-languages of Petri nets is the ordinal γ_21\gamma\_2^1, which is strictly greater than the first non-recursive ordinal ω_1CK\omega\_1^{{\rm CK}}. We also prove that there are some Σ_11{\bf \Sigma}\_1^1-complete, hence non-Borel, ω\omega-languages of Petri nets, and that it is consistent with ZFC that there exist some ω\omega-languages of Petri nets which are neither Borel nor Σ_11{\bf \Sigma}\_1^1-complete. This answers the question of the topological complexity of ω\omega-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

    Borel Ranks and Wadge Degrees of Context Free Omega Languages

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    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]

    Polishness of some topologies related to word or tree automata

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    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

    An Effective Extension of the Wagner Hierarchy to Blind Counter Automata

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    International audienceThe extension of the Wagner hierarchy to blind counter automata accepting infinite words with a Muller acceptance condition is effective. We determine precisely this hierarchy

    The Determinacy of Context-Free Games

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    We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter B\"uchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of omega-languages accepted by 1-counter B\"uchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter B\"uchi automaton A and a B\"uchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(A), L(B)).Comment: To appear in the Proceedings of the 29 th International Symposium on Theoretical Aspects of Computer Science, STACS 201

    08271 Abstracts Collection -- Topological and Game-Theoretic Aspects of Infinite Computations

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    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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