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

2001
*
Journal of Symbolic Logic (JSL)
*

every

doi:10.2307/2694911
fatcat:6vlrdpclkzajhjxcsdt6c6vbwa
*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. ...##
###
Topological Complexity of omega-Powers : Extended Abstract
[article]

2008
*
arXiv
*
pre-print

This is an extended abstract presenting new results on the topological complexity

arXiv:0809.1812v1
fatcat:v64qfyhasvhwdoe72lc3ubtp6e
*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 ...##
###
An upper bound on the complexity of recognizable tree languages

2015
*
RAIRO - Theoretical Informatics and Applications
*

Next, using an embedding

doi:10.1051/ita/2015002
fatcat:me234knrwrcwdhzhde6zjsosoy
*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 ...##
###
An Upper Bound on the Complexity of Recognizable Tree Languages
[article]

2015
*
arXiv
*
pre-print

Next, using an embedding

arXiv:1503.02840v2
fatcat:fpifejyktvbqvixcedk6ishtqe
*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 ...##
###
Borel ranks and Wadge degrees of context free $\omega$-languages

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. ...

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

2007
*
arXiv
*
pre-print

the supremum

arXiv:0712.1359v1
fatcat:neek5lsw3zctbfsblzevl3vbvq
*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. ...##
###
Wadge Degrees of ω-Languages of Petri Nets
[article]

2018
*
arXiv
*
pre-print

We prove that ω-languages

arXiv:1712.07945v2
fatcat:t52che3a4rdqlny52ystohfrvi
*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 . ...##
###
On the High Complexity of Petri Nets $$\omega $$-Languages
[chapter]

2020
*
Lecture Notes in Computer Science
*

ω-languages

doi:10.1007/978-3-030-51831-8_4
fatcat:3kxxriteq5ewropgozkc2vfnmu
*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 ...##
###
JSL volume 66 issue 1 Cover and Front matter

2001
*
Journal of Symbolic Logic (JSL)
*

Business correspondence should be sent to the Secretary-Treasurer

doi:10.1017/s002248120001118x
fatcat:fbasi4asg5dcre2lodhlrdssou
*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." ...##
###
On the expressive power of non-deterministic and unambiguous Petri nets over infinite words
[article]

2021
*
arXiv
*
pre-print

ω-languages

arXiv:2107.04025v2
fatcat:3szkjxamh5funcbqjdzpsd2qo4
*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. ...##
###
A syntactic approach to Borel functions: Some extensions of Louveau's theorem
[article]

2021
*
arXiv
*
pre-print

Louveau showed that if a

arXiv:2103.02950v1
fatcat:q6khftebuzbwtaww2c4bmwvkia
*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*. ...##
###
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 ...

##
###
Topological Complexity of omega-Powers: Extended Abstract

2008

On the other hand, the

doi:10.4230/dagsemproc.08271.7
fatcat:7zqpyw4zvreb5etack7k6zo6nq
*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 ...##
###
On Unambiguous Regular Tree Languages of Index (0,2)

unpublished

height

fatcat:hc4y2fqtgfcmbit2ey6lqgckeu
*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*. ...##
###
On some topics around the Wadge rank $ω_2$
[article]

2022

Kechris

doi:10.48550/arxiv.2201.12472
fatcat:nlqrxe6cxff5fbsa65fjikqpum
*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. ...
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