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From Automatic Structures to Borel Structures

Greg Hjorth, Bakh Khoussainov, Antonio Montalbán, André Nies
2008 Logic in Computer Science  
To answer these questions we introduce Borel structures and use some of the basic properties of Borel sets and isomorphisms.  ...  We study the classes of Büchi and Rabin automatic structures.  ...  In order to investigate an automatic version of the Löwenheim-Skolem theorem, we study Büchi and Rabin automatic structures from a finitistic view point.  ... 
doi:10.1109/lics.2008.28 dblp:conf/lics/HjorthKMN08 fatcat:cko4fcoz3jczzkpfb4kyd462zq

Polish topologies on groups of non-singular transformations [article]

François Le Maître
2022 arXiv   pre-print
We then characterize the Borel σ-finite measures λ on a standard Borel space for which the group of λ-preserving transformations has the automatic continuity property.  ...  Later on Sabok developed a framework to show automatic continuity for automorphism groups of metric structures [Sab19] .  ...  A characterization of automatic continuity We finally use the previous results to characterize automatic continuity for Aut(Y, λ), where (Y, λ) be a standard Borel space equipped with a Borel σ-finite  ... 
arXiv:2110.07289v2 fatcat:r3nevgwijjcircve5qhrxs7cga

Borel structures and Borel theories

Greg Hjorth, André Nies
2011 Journal of Symbolic Logic (JSL)  
We also investigate Borel isomorphisms between Borel structures.  ...  We show that there is a complete, consistent Borel theory which has no "Borel model" in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are  ...  We thank Bakhadyr Khoussainov for his suggestion to include Borel dimension, and Antonio Montalbán for helpful discussions. We also thank the referee for suggestions on how to improve the paper.  ... 
doi:10.2178/jsl/1305810759 fatcat:grsdvmj6sfc5zo5nqdfxbfq7qy

The isomorphism relation between tree-automatic Structures

Olivier Finkel, Stevo Todorčević
2010 Central European Journal of Mathematics  
An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata.  ...  We investigate in this paper the isomorphism problem for ω-tree-automatic structures.  ...  We are indebted to Anatole Khélif who was generously sharing with us his expertise about unitriangular groups and to the anonymous referees for very useful comments on a preliminary version of this paper  ... 
doi:10.2478/s11533-010-0014-7 fatcat:kkwhax72szc55dsey7xzeehcbi

Page 659 of Mathematical Reviews Vol. 25, Issue 4 [page]

1963 Mathematical Reviews  
W® has a natural Borel structure which makes it Borel- isomorphic to a Borel subset of the unit interval. Theorem 1 of this paper (which the author attributes to J.  ...  a natural Borel structure (which, the author shows, separates points).  ... 

Local Coloring Problems on Smooth Graphs [article]

Anton Bernshteyn
2021 arXiv   pre-print
We construct a smooth locally finite Borel graph G and a local coloring problem Π such that G has a coloring V(G) →ℕ that solves Π, but no such coloring can be Borel.  ...  Such a function f is called a Borel reduction from B to A.  ...  Re mov i n g t h e ex t ra s t r u c t u re In this section we sketch how the construction from §2 can be modified to make G a "plain" graph without any extra structure.  ... 
arXiv:2012.11031v2 fatcat:2cifdzo6inh73curbtjxhtw2ke

Invariant universality for projective planes [article]

Filippo Calderoni, Gianluca Paolini
2020 arXiv   pre-print
We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly  ...  We use constructions of the second author to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic.  ...  Aut(T ) ∼ = Aut(f (T ))), then clearly condition (II) from Fact 14 is automatically satisfied. Planes Definition 16. Let P 1 and P 2 be planes (cf.  ... 
arXiv:1801.10107v3 fatcat:qj3jqgncw5fyphqnx2e5gsrvfu

Structurable equivalence relations

Ruiyuan Chen, Alexander S. Kechris
2018 Fundamenta Mathematicae  
For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class.  ...  We show that K-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations.  ...  We need to review some standard concepts from the theory of Borel equivalence relations.  ... 
doi:10.4064/fm428-7-2017 fatcat:3wv3gcyaq5gupp7v5yw3464emq

Three Lectures on Automatic Structures [article]

Bakhadyr Khoussainov, Mia Minnes
2008 arXiv   pre-print
We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Buchi automata, and Rabin automata.  ...  The study of properties of automatic structures is a relatively new and very active area of research.  ...  We conclude by looking at Borel structures from descriptive set theory and connecting them to Büchi and Rabin automatic structures. 1 Basics Motivating questions The study of structures has played a  ... 
arXiv:0809.3430v1 fatcat:s5o3fh45bbf3dmo6ms3h6pjiia

Structurable equivalence relations [article]

Ruiyuan Chen, Alexander S. Kechris
2018 arXiv   pre-print
For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class.  ...  We show that K-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations.  ...  We would like to thank Andrew Marks for many valuable suggestions and for allowing us to include Theorem 1.11 in this paper.  ... 
arXiv:1606.01995v5 fatcat:b5qy5eim4fgp5jjcplmepwgrgi

Progress of robotized systems for crystallography for beamlines and laboratories

Jean-Luc Ferrer, Xavier Vernede, Yoann Sallaz-Damaz, Christophe Berzin, Michel Pirocchi, Franck Borel, Nathalie Larive
2014 Acta Crystallographica Section A: Foundations and Advances  
Thanks to its tool changer, the robot arm can go automatically from one application to another. G-Rob can be easily upgraded with new functions.  ...  From the seminal work accomplished on beamline FIP-BM30A (ESRF) in 2000' to the present developments, robot based systems significantly changed the crystallography experiment strategy.  ...  Thanks to its tool changer, the robot arm can go automatically from one application to another. G-Rob can be easily upgraded with new functions.  ... 
doi:10.1107/s2053273314092110 fatcat:xycqp4k3rrcoxkl67gonw4usda

Representations of étale groupoids on L^p-spaces [article]

Eusebio Gardella, Martino Lupini
2017 arXiv   pre-print
Our results yield new information about these objects: their matricially normed structure is uniquely determined.  ...  As a consequence of our main result, we deduce that every contractive representation of F^p(G) or F_red^p(G) is automatically completely contractive.  ...  It follows from the definition of the Borel structures on Z and Z ′ , that the canonical pairing Z * Z ′ → C is Borel.  ... 
arXiv:1408.3752v3 fatcat:6imrgjdsxzb4zju666ihegpdju

Coarse groups, and the isomorphism problem for oligomorphic groups [article]

Andre Nies, Philipp Schlicht, Katrin Tent
2022 arXiv   pre-print
In particular we can recover an isomorphic copy of G from ℳ(G) in a Borel fashion.  ...  We use the concept of a coarse group to show that the isomorphism relation for oligomorphic subgroups of S_∞ is Borel reducible to a Borel equivalence relation with all classes countable.  ...  They provided a Borel map M from the set of Roelcke precompact closed subgroups of S ∞ to structures in X L ; we think of the domain of the structure M(G) as consisting of the cosets of subgroups in N  ... 
arXiv:1903.08436v3 fatcat:lqmxfqxxqjdg3piokte577tfzy

The classification problem for finitely generated operator systems and spaces [article]

Martín Argerami, Samuel Coskey, Mehrdad Kalantar, Matthew Kennedy, Martino Lupini, Marcin Sabok
2015 arXiv   pre-print
The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory.  ...  show that the finitely generated operator systems and spaces are completely classifiable (or smooth); in fact a finitely generated operator system is classified by its complete theory when regarded as a structure  ...  By Proposition 5.3 the function M → ϕ M n n∈N from Mod p (L) to R N is a Borel reduction from the relation of isomorphism of proper L-structures to equality of sequences of real numbers.  ... 
arXiv:1411.0512v3 fatcat:vzw556d7fncu5oggvko7lctubm

Polish groupoids and functorial complexity [article]

Martino Lupini
2014 arXiv   pre-print
In order to obtain such a conclusion we generalize some fundamental results about the descriptive set theory of Polish group actions to actions of Polish groupoids, answering a question of Arlan Ramsay  ...  These include the Becker-Kechris results on Polishability of Borel universal Borel G-spaces, and characterization of Borel G-spaces with Borel orbit equivalence relations.  ...  G 0 to the space F (G 0 ) of closed subsets of x endowed with the Effros Borel structure is a reduction from E X G to equality in G 0 .  ... 
arXiv:1407.6671v2 fatcat:knc7l52yovewtnx7ha6urop4qa
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