1,798 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

    Total Representations

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    Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.Comment: 30 page
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