On the existence of universal numberings for finite families of d.c.e. sets

Abeshev, Kuanysh

doi:10.1002/malq.201200045

Cited by 6 publications

(7 citation statements)

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“…The study of Rogers semilattices in the Ershov hierarchy is interesting because in it a er of unexpected results have been obtained. For example, it was shown in [15] that there is ily 𝑆𝑆 consisting of just two 𝑑𝑑-c.e. sets such that 𝐶𝐶𝐶𝐶𝑚𝑚 2 −1 (𝑆𝑆) has no universal numbering, -computable (or equivalently, n-computable), if the relation y.…”

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

“…y of Rogers semilattices in the Ershov hierarchy is interesting because in it a pected results have been obtained. For example, it was shown in [15] that there is isting of just two 𝑑𝑑-c.e. sets such that 𝐶𝐶𝐶𝐶𝑚𝑚 2 −1 (𝑆𝑆) has no universal numbering, is in all here that a set 𝐴𝐴 ⊆ 𝜔𝜔 is in Ershov's hierarchy class Σ 𝑛𝑛 −1 if A is 𝑛𝑛le (𝑛𝑛-c.e.…”

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

“…The study of Rogers semilattices in the Ershov hierarchy is interesting because in it a number of unexpected results have been obtained. For example, it was shown in [15] that there is…”

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

“…lattices in the Ershov hierarchy is interesting because in it a been obtained. For example, it was shown in [15] that there is 𝑑𝑑-c.e. sets such that 𝐶𝐶𝐶𝐶𝑚𝑚 −1 (𝑆𝑆) has no universal numbering, modulo equivalence of the numberings ordered by the relation reducibility of numberings is also called Rogers semilattice and denote it as Ershov hierarchy.…”

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

“…The study of Rogers semilattices in the Ershov hierarchy is interesting because in it a number of unexpected results have been obtained. For example, it was shown in [15] that there is a family 𝑆𝑆 consisting of just two 𝑑𝑑-c.e. sets such that 𝐶𝐶𝐶𝐶𝑚𝑚 2 −1 (𝑆𝑆) has no universal numbering, despite the fact that in classical numbering theory every finite family has a universal numbering.…”

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

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On the Existence of Universal Numberings

Nurlanbek

2023

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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: Main Provisions Materials and Methodsmentioning

confidence: 99%

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

“…The study of Rogers semilattices in the Ershov hierarchy is interesting because in it a number of unexpected results have been obtained. For example, it was shown in [15] that there is…”

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

Section: Main Provisions Materials and Methodsmentioning

confidence: 99%

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