Degrees of bi-embeddable categoricity of equivalence structures

Bazhenov, Nikolay; Fokina, Ekaterina B.; Rossegger, Dino; Mauro, Luca San

doi:10.1007/s00153-018-0650-3

Cited by 8 publications

(4 citation statements)

References 30 publications

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“…They studied the complexity of embeddings structures. To facilitate this study, they introduced computable bi‐embeddable categoricity and classified the degrees of computable bi‐embeddable categoricity for equivalence structures .…”

Section: Introductionmentioning

confidence: 99%

“…They studied the complexity of embeddings structures. To facilitate this study, they introduced computable bi-embeddable categoricity and classified the degrees of computable bi-embeddable categoricity for equivalence structures [3,4]. The focus of this paper is the degree spectrum of A under bi-embeddability, or for short bi-embeddability spectrum of A,…”

Section: Introductionmentioning

confidence: 99%

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Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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We study degree spectra of structures with respect to the bi‐embeddability relation. The bi‐embeddability spectrum of a structure is the family of Turing degrees of its bi‐embeddable copies. To facilitate our study we introduce the notions of bi‐embeddable triviality and basis of a spectrum. Using bi‐embeddable triviality we show that several known families of degrees are bi‐embeddability spectra of structures. We then characterize the bi‐embeddability spectra of linear orderings and study bases of bi‐embeddability spectra of strongly locally finite graphs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bi‐embeddability spectra and bases of spectra

Fokina

Rossegger

Mauro

2019

Mathematical Logic Qtrly

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“…In particular, bi-embeddability is of fundamental interest for classifying the complexity of equivalence relations in terms of Borel reducibility, as in [8], and the effective countepart of Borel reducibility discussed, e.g., in [7]. See also [4] for a full classification of computable presentations of equivalence structures up to bi-embeddability.…”

Section: Introductionmentioning

confidence: 99%

Limit Learning Equivalence Structures

Fokina,

Kötzing,

Mauro

2019

Preprint

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While most research in Gold-style learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pairs of elements of a structure are related and which are not. The aim of the learner is to find (an effective description of) the isomorphism type of the structure presented in the limit. In accordance with language learning we call this learning criterion Inf Ex-learning (explanatory learning from informant).Our main contribution is a complete characterization of which families of equivalence structures are Inf Ex-learnable. This characterization allows us to derive a bound of 0 ′′ on the computational complexity required to learn uniformly enumerable families of equivalence structures. We also investigate variants of Inf Ex-learning, including learning from text (where the only information provided is which elements are related, and not which elements are not related) and finite learning (where the first actual conjecture of the learner has to be correct). Finally, we show how learning families of structures relates to learning classes of languages by mapping learning tasks for structures to equivalent learning tasks for languages.

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“…THEOREM 2 [9]. Any computable equivalence structure has degree of bi-embeddable categoricity d ∈ {0, 0 , 0 }.…”

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Computable Bi-Embeddable Categoricity

et al. 2018

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We study the algorithmic complexity of isomorphic embeddings between computable structures. Suppose that L is a language. We say that L-structures A and B are bi-embeddable (denoted A ≈ B) if there are isomorphic embeddings f : A → B and g : B → A. The systematic investigation of the bi-embeddability relation in computable structure theory was initiated by Montalbán [1, 2]: he proved that any hyperarithmetical linear order is bi-embeddable with a computable one. In [3], similar results were obtained for Abelian p-groups, Boolean algebras, and compact metric spaces. The paper [4] studies degree spectra with respect to bi-embeddability.Definition 1. Let d be a Turing degree. We say that a computable structure S is d-computably bi-embeddably categorical if, for any computable structure A ≈ S, there are d-computable isomorphic embeddings f : A → S and g : S → A. The bi-embeddable categoricity spectrum of S is the set CatSpec ≈ (S) = {d : S is d-computably bi-embeddably categorical}.

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Degrees of bi-embeddable categoricity of equivalence structures

Cited by 8 publications

References 30 publications

Bi‐embeddability spectra and bases of spectra

Bi‐embeddability spectra and bases of spectra

Limit Learning Equivalence Structures

Computable Bi-Embeddable Categoricity

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