A Survey on Universal Computably Enumerable Equivalence Relations

Andrews, Uri; Badaev, S. A.; Sorbi, Andrea

doi:10.1007/978-3-319-50062-1_25

Cited by 39 publications

(40 citation statements)

References 22 publications

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“…This result was later strengthened by Lachlan [18] (see also [1, p425]), who showed that all precomplete ceers are computably isomorphic. For a recent survey about ceers we refer the reader to Andrews, Badaev, and Sorbi [1].…”

Section: Numberings and Equivalence Relationsmentioning

confidence: 99%

Fixed point theorems for precomplete numberings

Barendregt

Terwijn

2019

Annals of Pure and Applied Logic

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In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.

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Section: Numberings and Equivalence Relationsmentioning

confidence: 99%

Fixed point theorems for precomplete numberings

Barendregt

Terwijn

2019

Annals of Pure and Applied Logic

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“…equivalence relations are often called positive, but as in [20] and many other contributions, we adopt the acronym ceers for referring to them. The interested reader can see Andrews, Badaev, and Sorbi [1] for a nice and up-to-date survey on ceers, with a special focus on universal ceers, i.e., ceers to which all others can be computably reduced.…”

Section: Introductionmentioning

confidence: 99%

“…sequence of numbers such that, for all i, y i ∈ W f (i) , and let Y = { k, y k : k ∈ ω}.Notice that the set Y * = {j : (∃x)( j, x ∈ h[Y ])} must be infinite. Otherwise, since each column of S is finitely dimensional, h would map some noncomputable equivalence class of R into a singleton of S, and therefore we would be in Case(1). Moreover, Y * ⊆ B.…”

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confidence: 99%

Measuring the complexity of reductions between equivalence relations

Fokina

Rossegger

Mauro

2019

COM

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Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.

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“…Let a be a notation for a computable ordinal. A set A Ď ω of numbers is said to be Σ´1 a (or A P Σ´1 a ) if there are computable functions f pz, tq and γpz, tq such that, for all z, (1) Apzq " lim t f pz, tq, with f pz, 0q " 0;…”

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confidence: 99%

Classifying equivalence relations in the Ershov hierarchy

et al. 2020

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Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility ďc. This gives rise to a rich degree-structure. In this paper, we lift the study of c-degrees to the ∆ 0 2 case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by ďc on the Σ´1 a Π´1 a equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

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A Survey on Universal Computably Enumerable Equivalence Relations

Cited by 39 publications

References 22 publications

Fixed point theorems for precomplete numberings

Fixed point theorems for precomplete numberings

Measuring the complexity of reductions between equivalence relations

Classifying equivalence relations in the Ershov hierarchy

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