Positive undecidable numberings in the Ershov hierarchy

Mustafa, Manat; Sorbi, Andrea

doi:10.1007/s10469-012-9162-0

Cited by 7 publications

(2 citation statements)

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“…In view of the properties of the F. Stephan operator [17], it suffices to research Rogers semilattices for families of sets at two lower levels in the Ershov hierarchy. Other results on Rogers semilattices in Ershov hierarchy can be found, for example, in [18][19][20][21][22][23][24][25][26].…”

Section: математические наукиmentioning

confidence: 99%

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On the Existence of Universal Numberings

Nurlanbek

2023

jour

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The paper is devoted to research existence property of universal numberings for different computable families. A numbering α is reducible to a numbering β if there is computable function ƒ such that α = β ◦ ƒ. A computable numbering α for some family S is universal if any computable numbering β for the family S is reducible to α. It is well known that the family of all computably enumerable (c.e.) sets has a computable universal numbering. In this paper, we study families of almost all c.e. sets, recursive sets, and almost all differences of c.e. sets, namely questions about the existence of universal numberings for given families. We proved that there is no universal numbering for the family of all recursive sets. For families of c.e. sets without an empty set or a finite number of finite sets, there still exists a universal numbering. However, for families of all c.e. sets without an infinite set, there is no universal numbering. Also, we proved that family ∑2-1 \ Β and the family ∑1-1 has no universal ∑2-1-computable numbering for any Β ∈ ∑2-1.

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Section: математические наукиmentioning

confidence: 99%

“…In the F. Stephan operator [17], it suffices to research Rogers semilattices f lower levels in the Ershov hierarchy. Other results on Rogers semilatt can be found, for example, in [18][19][20][21][22][23][24][25][26].…”

Section: математические наукиmentioning

confidence: 99%

On the Existence of Universal Numberings

Nurlanbek

2023

jour

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Friedberg numberings in the Ershov hierarchy

2014

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The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

Bazhenov

2015

Algebra Logic

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Computable categoricity in the Ershov hierarchy is studied. We consider F a -categorical and G a -categorical structures. These were introduced by B. Khoussainov, F. Stephan, and Y. Yang for a, which is a notation for a constructive ordinal. A generalization of the branching theorem is proved for F a -categorical structures. As a consequence we obtain a description of F a -categorical structures for classes of Boolean algebras and Abelian p-groups. Furthermore, it is shown that the branching theorem cannot be generalized to G a -categorical structures.Research on computable categoricity of structures was initiated by B. L. van der Waerden who studied the question whether there exists an effective algorithm for constructing an isomorphism between two different algebraic closures of a field built in different ways. Subsequently, starting with 1950s, computable categoricity of structures was explored by A. I. Mal'tsev, A. Fröhlich, J. Shepherdson, S. S. Goncharov, J. Remmel, C. Ash, J. Knight, and many other authors.In investigating computably categorical structures, one of the most important problems is to gain an insight into computable categoricity relative to classes of known hierarchies of sets. In particular, of special interest is the Ershov hierarchy introduced in [1-3]. There are quite many works dealing with the Ershov hierarchy. (Among these, it is worth mentioning resent papers [4-10] that focus primarily on numberings in the Ershov hierarchy.) Research on computable categoricity relative to levels of the Ershov hierarchy was started in [11]. There, for an arbitrary a, which is *

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Positive undecidable numberings in the Ershov hierarchy

Cited by 7 publications

References 4 publications

On the Existence of Universal Numberings

On the Existence of Universal Numberings

Friedberg numberings in the Ershov hierarchy

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

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