IA Scholar Query: Wadge Hierarchy and Veblen Hierarchy Part I: Borel Sets of Finite Rank.
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Internet Archive Scholar query results feedeninfo@archive.orgSat, 01 Jan 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On some topics around the Wadge rank $ω_2$
https://scholar.archive.org/work/ojyvb2dtmzg4dc7wfkjcqnqo4a
Kechris and Martin showed that the Wadge rank of the $ω$-th level of the decreasing difference hierarchy of coanalytic sets is $ω_2$ under the axiom of determinacy. In this article, we give an alternative proof of the Kechris-Martin theorem, by understanding the $ω$-th level of the decreasing difference hierarchy of coanalytic sets as the (relative) hyperarithmetical processes with finite mind-changes. Based on this viewpiont, we also examine the gap between the increasing and decreasing difference hierarchies of coanalytic sets by relating them to the $Π^1_1$- and $Σ^1_1$-least number principles, respectively. We also analyze Weihrauch degrees of related principles.Takayuki Kiharawork_ojyvb2dtmzg4dc7wfkjcqnqo4aSat, 01 Jan 2022 00:00:00 GMTOn the expressive power of non-deterministic and unambiguous Petri nets over infinite words
https://scholar.archive.org/work/6veutlophvdp7ibw3dyove4thi
We prove that ω-languages of (non-deterministic) Petri nets and ω-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Turing machines. We also show that it is highly undecidable to determine the topological complexity of a Petri net ω-language. Moreover, we infer from the proofs of the above results that the equivalence and the inclusion problems for ω-languages of Petri nets are Π_2^1-complete, hence also highly undecidable. Additionally, we show that the situation is quite the opposite when considering unambiguous Petri nets, which have the semantic property that at most one accepting run exists on every input. We provide a procedure of determinising them into deterministic Muller counter machines with counter copying. As a consequence, we entail that the ω-languages recognisable by unambiguous Petri nets are Δ^0_3 sets.Olivier Finkel, Michał Skrzypczakwork_6veutlophvdp7ibw3dyove4thiTue, 14 Dec 2021 00:00:00 GMT