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Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank

J. Duparc
2001 Journal of Symbolic Logic (JSL)  
every Borel set A of finite rank to its Wadge degree.  ...  We consider Borel sets of finite rank A ⊆ ∧ ω where cardinality of Λ is less than some uncountable regular cardinal .  ...  Thus theorem 38 provides for every Borel set of finite rank A a normal form Q Wadge equivalent to A.  ... 
doi:10.2307/2694911 fatcat:6vlrdpclkzajhjxcsdt6c6vbwa

Topological Complexity of omega-Powers : Extended Abstract [article]

Olivier Finkel
2008 arXiv   pre-print
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.  ...  The Wadge hierarchy of Borel sets of finite rank has length 1 ε 0 where 1 ε 0 is the limit of the ordinals α n defined by α 1 = ω 1 and α n+1 = ω αn 1 for n a non negative integer, ω 1 being the first  ... 
arXiv:0809.1812v1 fatcat:v64qfyhasvhwdoe72lc3ubtp6e

An upper bound on the complexity of recognizable tree languages

Olivier Finkel, Dominique Lecomte, Pierre Simonnet
2015 RAIRO - Theoretical Informatics and Applications  
Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space 2 ω into the class ∆ 1 2 , and the notions of Wadge degree and Veblen function, we argue that this upper  ...  that every regular tree language of infinite trees is in a class (D n (Σ 0 2 )) for some natural number n ≥ 1, where is the game quantifier. We first give a detailed exposition of this result.  ...  We wish to thank the anonymous referee for very useful comments on a preliminary version of this paper which lead to a great improvement of the paper, and for indicating us some related studies that are  ... 
doi:10.1051/ita/2015002 fatcat:me234knrwrcwdhzhde6zjsosoy

An Upper Bound on the Complexity of Recognizable Tree Languages [article]

Olivier Finkel, Pierre Simonnet
2015 arXiv   pre-print
Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space 2^ω into the class Δ^1_2, and the notions of Wadge degree and Veblen function, we argue that this upper  ...  We first give a detailed exposition of this result.  ...  We wish to thank the anonymous referee for very useful comments on a preliminary version of this paper which lead to a great improvement of the paper, and for indicating us some related studies that are  ... 
arXiv:1503.02840v2 fatcat:fpifejyktvbqvixcedk6ishtqe

Borel ranks and Wadge degrees of context free $\omega$-languages

OLIVIER FINKEL
2006 Mathematical Structures in Computer Science  
And the supremum of the set of Borel ranks of context free ω-languages is an ordinal γ 1 2 which is strictly greater than the first non recursive ordinal ω CK 1 .  ...  We show that the Borel hierarchy of the class of context free ω-languages, or even of the class of ω-languages accepted by Büchi 1-counter automata, is the same as the Borel hierarchy of the class of ω-languages  ...  Acknowledgements Thanks to the anonymous referee for useful comments on a preliminary version of this paper.  ... 
doi:10.1017/s0960129506005597 fatcat:g3jri3nlt5aurozzy5neishngy

Borel Ranks and Wadge Degrees of Context Free Omega Languages [article]

Olivier Finkel
2007 arXiv   pre-print
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.  ...  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  ...  Acknowledgements Thanks to the anonymous referee for useful comments on a preliminary version of this paper.  ... 
arXiv:0712.1359v1 fatcat:neek5lsw3zctbfsblzevl3vbvq

Wadge Degrees of ω-Languages of Petri Nets [article]

Olivier Finkel
2018 arXiv   pre-print
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  ...  In particular, for each non-null recursive ordinal α < ω_1^ CK there exist some Σ^0_α-complete and some Π^0_α-complete ω-languages of Petri nets, and the supremum of the set of Borel ranks of ω-languages  ...  The Wadge hierarchy W H is the class of Borel subsets of a set X ω , where X is a finite set, equipped with ≤ W and with ≡ W .  ... 
arXiv:1712.07945v2 fatcat:t52che3a4rdqlny52ystohfrvi

On the High Complexity of Petri Nets $$\omega $$-Languages [chapter]

Olivier Finkel
2020 Lecture Notes in Computer Science  
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 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  ...  The Wadge hierarchy of Borel sets of finite rank has length 1 ε 0 where 1 ε 0 is the limit of the ordinals α n defined by α 1 = ω 1 and α n+1 = ω αn 1 for n a non negative integer, ω 1 being the first  ... 
doi:10.1007/978-3-030-51831-8_4 fatcat:3kxxriteq5ewropgozkc2vfnmu

JSL volume 66 issue 1 Cover and Front matter

2001 Journal of Symbolic Logic (JSL)  
Business correspondence should be sent to the Secretary-Treasurer of the Association, Charles Steinhorn (address on inside back cover). k _ 1 Wadge hierarchy and Veblen hierarchy Part I: Borel sets  ...  of finite rank,,by J."  ... 
doi:10.1017/s002248120001118x fatcat:fbasi4asg5dcre2lodhlrdssou

On the expressive power of non-deterministic and unambiguous Petri nets over infinite words [article]

Olivier Finkel, Michał Skrzypczak
2021 arXiv   pre-print
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 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  ...  The authors also thank anonymous referees of this paper and of the papers incorporated in this journal version -their suggestions allowed to improve the presentation.  ... 
arXiv:2107.04025v2 fatcat:3szkjxamh5funcbqjdzpsd2qo4

A syntactic approach to Borel functions: Some extensions of Louveau's theorem [article]

Takayuki Kihara, Kenta Sasaki
2021 arXiv   pre-print
Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class Γ, then its Γ-code can be obtained from its Borel code in a hyperarithmetical manner.  ...  More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions.  ...  The notion of Wadge degrees [37] provides us an ultimate refinement of all known hierarchies in descriptive set theory, such as the Borel hierarchy and the Hausdorff-Kuratowski difference hierarchy.  ... 
arXiv:2103.02950v1 fatcat:q6khftebuzbwtaww2c4bmwvkia

JSL volume 66 issue 4 Cover and Back matter

2001 Journal of Symbolic Logic (JSL)  
Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank 56 EHRLICH, PHILIP.  ...  -adic fields 1493 BUECHLER, STEVEN and HOOVER, COLLEEN. The classification of small types of rank co, Part I 1884 BUNDER, MARTIN and DEKKERS, WIL.  ...  The JOURNAL and the BULLETIN OF SYMBOLIC LOGIC are typeset in a version of A_MS-1£T]*X. using a stylefile which is posted on the ASL Website, h t t p : //www. a s l o n l i n e . o r g , along with instructions  ... 
doi:10.1017/s0022481200010124 fatcat:dvgtmcubczd6nm5nvjjz6tvjnq

Topological Complexity of omega-Powers: Extended Abstract

Olivier Finkel, Dominique Lecomte, Peter Hertling, Victor Selivanov, Wolfgang Thomas, William W. Wadge, Klaus Wagner
2008
On the other hand, the Wadge hierarchy is a great refinement of the Borel hierarchy, determined by Bill Wadge.  ...  We show that, for each ordinal $xi$ greater than or equal to 3, there are uncountably many Wadge degrees of omega-powers of Borel rank $xi +1$.  ...  The Wadge hierarchy W H is the class of Borel subsets of a set X ω , where X is a finite set, equipped with ≤ W and with ≡ W  ... 
doi:10.4230/dagsemproc.08271.7 fatcat:7zqpyw4zvreb5etack7k6zo6nq

On Unambiguous Regular Tree Languages of Index (0,2)

Jacques Duparc, Kevin Fournier, Szczepan Hummel
unpublished
height of the Wadge hierarchy of deterministic tree languages as uncovered by Filip Murlak. 3.  ...  We do so by providing operations that generate a family A b α α<ϕ 2 (0) of unambiguous automata such that: 1. It respects the strict Wadge ordering: α < β if and only if A b α < W A b β.  ...  In particular, the Wadge hierarchy refines tremendously the Borel and the Projective hierarchies.  ... 
fatcat:hc4y2fqtgfcmbit2ey6lqgckeu

On some topics around the Wadge rank $ω_2$ [article]

Takayuki Kihara
2022
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.  ...  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  ...  The author is very grateful to Kenta Sasaki for thorough discussions and valuable comments.  ... 
doi:10.48550/arxiv.2201.12472 fatcat:nlqrxe6cxff5fbsa65fjikqpum
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