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The Borel hierarchy is infinite in the class of regular sets of trees

Jerzy Skurczyński
1993 Theoretical Computer Science  
In this situation at least two natural questions arise: (1) Is the Bore1 hierarchy infinite in the class of regular sets of trees or does it collapse at some level?  ...  In the case of regular sets of infinite trees the full topological classification is still unknown (according to the author's knowledge).  ...  For t$C, we can always find such t'EC that d(t, t')=minrscd(t, t) (because C is closed). Let T = (w, T) be defined by the formula We take q(t)=z.  ... 
doi:10.1016/0304-3975(93)90030-w fatcat:rxeho5w22vblld5lbyed2cz7zu

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

Olivier Finkel, Pierre Simonnet
2015 arXiv   pre-print
The third author noticed in his 1992 PhD Thesis [Sim92] 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.  ...  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  ...  now present in the bibliography.  ... 
arXiv:1503.02840v2 fatcat:fpifejyktvbqvixcedk6ishtqe

An upper bound on the complexity of recognizable tree languages

Olivier Finkel, Dominique Lecomte, Pierre Simonnet
2015 RAIRO - Theoretical Informatics and Applications  
The third author noticed in his 1992 PhD Thesis [?] 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.  ...  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  ...  now present in the bibliography.  ... 
doi:10.1051/ita/2015002 fatcat:me234knrwrcwdhzhde6zjsosoy

On Recognizable Tree Languages Beyond the Borel Hierarchy [article]

Olivier Finkel
2009 arXiv   pre-print
We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets.  ...  In particular these languages W_(i, k) are not in any class D_α(Σ^1_1) for α < ω^ω.  ...  We thank the anonymous referees for their very helpful comments which have led to a great improvement of our paper.  ... 
arXiv:0909.0393v1 fatcat:zcnorxtnkbdxvh6ph5ckpoe3ua

Borel Hierarchy and Omega Context Free Languages [article]

Olivier Finkel
2011 arXiv   pre-print
And one cannot decide whether an omega-CFL is a Borel set.  ...  We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621], proving new  ...  We have previously proved the existence of analytic but non Borel sets in another class of ω-languages, the class of locally finite ωlanguages [Fin01c] .  ... 
arXiv:1101.3443v1 fatcat:mchn5kjacvautj2yks4psn5aqa

Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

Olivier Finkel
2003 RAIRO - Theoretical Informatics and Applications  
Then we show that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class Σ 0 α ( respectively Π 0 α ).  ...  In particular one cannot decide whether an infinitary rational relation is an arithmetical set.  ...  As in the case of the Borel hierarchy, projections of arithmetical sets (of the second Π-class) lead beyond the Arithmetical hierarchy, to the Analytical hierarchy of ω-languages.  ... 
doi:10.1051/ita:2003013 fatcat:m6x2mb6nwrdrje3ipn4v5lkxou

Regular Tree Languages in the First Two Levels of the Borel Hierarchy

Filippo Cavallari
2019 Bulletin of Symbolic Logic  
In this thesis we instead drop any restriction and prove some new results concerning arbitrary regular tree languages which belong to low levels of the Borel hierarchy and of the Wadge hierarchy.  ...  In Descriptive Set Theory, such subsets are usually stratified in topological hierarchies, like the Borel hierarchy, the Wadge hierarchy and the difference hierarchy; in Automata Theory, such sets are  ...  a differential polynomial equation defines a closed set in the Kolchin topology, the differential analogue of the Zariski topology.  ... 
doi:10.1017/bsl.2019.12 fatcat:aiclntnl7ncatisjx2y57hcdwi

On the Topological Complexity of Infinitary Rational Relations

Olivier Finkel
2003 RAIRO - Theoretical Informatics and Applications  
We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20] .  ...  The classes Σ 0 n and Π 0 n of the Borel Hierarchy on the topological space X ω are defined as follows: Σ 0 1 is the class of open sets of X ω ; Π 0 1 is the class of closed sets of X ω .  ...  A set L ⊆ X ω is a closed set iff its complement X ω − L is an open set. We define now the next classes of the Borel Hierarchy: Definition 3.1.  ... 
doi:10.1051/ita:2003016 fatcat:uxy4uhxa3zcylkmrptaubthvpq

Regular Languages of Thin Trees

Tomasz Idziaszek, Michał Skrzypczak, Mikołaj Bojańczyk
2015 Theory of Computing Systems  
Using this framework we characterize various classes of regular languages: commutative, open in the standard topology, closed under two variants of bisimulational equivalence, and definable in WMSO logic  ...  In this work we investigate properties of regular languages of thin trees. Our main tool is an algebra suitable for thin trees.  ...  Acknowledgements The authors would like to thank Henryk Michalewski for posing a number of motivating problems and questions on the subject.  ... 
doi:10.1007/s00224-014-9595-z fatcat:v7mrzp2xyzc6lndxemzdmkn3my

On the Topological Complexity of Infinitary Rational Relations [article]

Olivier Finkel
2008 arXiv   pre-print
We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Automates et Th\'eorie Descriptive, Ph.  ...  The classes Σ 0 n and Π 0 n of the Borel Hierarchy on the topological space X ω are defined as follows: Σ 0 1 is the class of open sets of X ω . Π 0 1 is the class of closed sets of X ω .  ...  A set L ⊆ X ω is a closed set iff its complement X ω − L is an open set. We define now the next classes of the Borel Hierarchy: Definition 3.1.  ... 
arXiv:0803.1841v1 fatcat:dnhjuvgedbghlkzdvtj3jj5bqi

Baire and automata

Pierre Simonnet, Benoit Cagnard
2007 Discrete Mathematics & Theoretical Computer Science  
International audience In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions.  ...  Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity.  ...  Acknowledgements Acknowledgements of the first author : I began to learn automata theory in the DEA of Maurice Nivat (my adviser) in the year 1983-1984 at LITP.  ... 
doi:10.46298/dmtcs.392 fatcat:a5m37czmonfwpi432y2jqylblu

On the Topological Complexity of MSO+U and Related Automata Models [chapter]

Szczepan Hummel, Michał Skrzypczak, Szymon Toruńczyk
2010 Lecture Notes in Computer Science  
We also give an exact topological complexity of the classes of languages recognized by nondeterministic ωB-, ωSand ωBS-automata studied by Bojańczyk and Colcombet in [BC06] .  ...  We show that Monadic Second Order Logic on ω-words extended with the unbounding quantifier (MSO+U) can define non-Borel sets.  ...  We would like to thank the anonymous referees for their constructive comments.  ... 
doi:10.1007/978-3-642-15155-2_38 fatcat:fozaa3dyfzavnallf5fsrcre34

Polishness of some topologies related to word or tree automata

Olivier Finkel, Olivier Carton, Dominique Lecomte
2019 Logical Methods in Computer Science  
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.  ...  We prove that the B\"uchi topology and the automatic topology are Polish.  ...  The second author thanks very much Henryk Michalewski who suggested to study the analogue of the Büchi topology in the case of a space of infinite labelled trees.  ... 
doi:10.23638/lmcs-15(2:9)2019 fatcat:ohmcgskxofeapma2dz7lqskrrq

Deciding the Borel Complexity of Regular Tree Languages [chapter]

Alessandro Facchini, Henryk Michalewski
2014 Lecture Notes in Computer Science  
We show that it is decidable whether a given a regular tree language belongs to the class ∆ 0 2 of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.  ...  The author is supported by the Expressiveness of Modal Fixpoint Logics project realized within the 5/2012 Homing Plus programme of the Foundation for Polish Science, co-financed by the European Union from  ...  the Regional Development Fund within the Operational Programme Innovative Economy ("Grants for Innovation")  ... 
doi:10.1007/978-3-319-08019-2_17 fatcat:2kxefdpwdngxfakx3bvhr7qsnm

Polishness of some topologies related to word or tree automata [article]

Olivier Finkel, Olivier Carton, Dominique Lecomte
2019 arXiv   pre-print
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.  ...  We prove that the B\"uchi topology and the automatic topology are Polish.  ...  The second author thanks very much Henryk Michalewski who suggested to study the analogue of the Büchi topology in the case of a space of infinite labelled trees.  ... 
arXiv:1710.04002v4 fatcat:zh57taz6vbaqrgctaziqs5p6ra
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