S-Subsets of Natural Numbers

Morozov, Andrey; Puzarenko, V. G.

doi:10.1023/b:allo.0000028930.44605.68

Cited by 28 publications

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“…Key properties of the model N A are the following [9]: Since M A Σ HF(N A ) and A is a Σ-subset of HF(M A ), B e A holds for any B ⊆ ω iff B is a Σ-subset of HF(M A ) (see [11]). …”

Section: Existence Of a Universal Functionmentioning

confidence: 99%

Descriptive properties on admissible sets

Puzarenko

2010

Algebra Logic

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Relations are treated between the following descriptive properties on admissible sets: enumerability, uniformization, reduction, separation, extension. Moreover, in the setting of these properties, we consider existence problems for a universal computable function and for a computable function universal for {0; 1}-valued computable functions. It is shown that all relations between the given properties are strict. Also we look into algorithmic complexity of admissible sets lending support to the specified relations. It is stated that the reduction principle fails in some admissible sets over classical structures.For the majority of approaches to computability, a class of computably enumerable sets, which, as a rule, form a lattice with respect to set-theoretic operations, is not closed under complements. (Moreover, elements that have complements are exactly computable sets.) This necessitates the study of additional set-theoretic properties of the given class. In the present paper, we look at such properties for admissible sets. It will be shown that admissible sets behave rather badly in relation to much used properties. Namely, every relation between the properties may be realized on a certain admissible set, and in most cases, such a set can be chosen as a hereditarily finite superstructure over a computable model.

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“…Key properties of the model N A are the following [9]: Since M A Σ HF(N A ) and A is a Σ-subset of HF(M A ), B e A holds for any B ⊆ ω iff B is a Σ-subset of HF(M A ) (see [11]). …”

Section: Existence Of a Universal Functionmentioning

confidence: 99%

Descriptive properties on admissible sets

Puzarenko

2010

Algebra Logic

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“…We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in HF(A) coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in HF(A) coincides with I.Problems of Σ-definability of subsets of the set of finite ordinals in admissible sets were addressed in articles [1][2][3][4][5]. Connections between T -reducibility and Σ-definability were studied in [2,3,5], and relations between e-reducibility and the family of Σ-subsets of ω in admissible sets, in [1,4].…”

mentioning

confidence: 99%

“…There are examples in [1] of the models in whose hereditarily finite extensions the family of Σ-definable subsets of ω coincides with I * = {S ⊆ ω | d e (S) ∈ I}, where I is an arbitrary e-ideal. The present article is inspired by [1].The necessary background on admissible sets can be found in [6,7]. The fundamentals of the classical computability theory and group theory can be obtained from [8] and [9] respectively.…”

mentioning

confidence: 99%

“…Problems of Σ-definability of subsets of the set of finite ordinals in admissible sets were addressed in articles [1][2][3][4][5]. Connections between T -reducibility and Σ-definability were studied in [2,3,5], and relations between e-reducibility and the family of Σ-subsets of ω in admissible sets, in [1,4].…”

mentioning

confidence: 99%

“…Connections between T -reducibility and Σ-definability were studied in [2,3,5], and relations between e-reducibility and the family of Σ-subsets of ω in admissible sets, in [1,4]. There are examples in [1] of the models in whose hereditarily finite extensions the family of Σ-definable subsets of ω coincides with I * = {S ⊆ ω | d e (S) ∈ I}, where I is an arbitrary e-ideal. The present article is inspired by [1].…”

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confidence: 99%

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On Σ-subsets of naturals over abelian groups

Khisamiev

2006

Sib Math J

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We obtain conditions for the Σ-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in HF(A) coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in HF(A) coincides with I.Problems of Σ-definability of subsets of the set of finite ordinals in admissible sets were addressed in articles [1][2][3][4][5]. Connections between T -reducibility and Σ-definability were studied in [2,3,5], and relations between e-reducibility and the family of Σ-subsets of ω in admissible sets, in [1,4]. There are examples in [1] of the models in whose hereditarily finite extensions the family of Σ-definable subsets of ω coincides with I * = {S ⊆ ω | d e (S) ∈ I}, where I is an arbitrary e-ideal. The present article is inspired by [1].The necessary background on admissible sets can be found in [6,7]. The fundamentals of the classical computability theory and group theory can be obtained from [8] and [9] respectively. In this article we consider hereditarily finite admissible sets over models of finite signatures.Our notation is standard. Denote by W n the nth computably enumerable set; and by D n , the nth finite set: D n = {a 1 , . . . , a k } for n = k i=1 2 a i . Enumeration reducibility, or e-reducibility for brevity, is defined byA ≤ e B ⇔ ∃n∀t (t ∈ A ⇔ ∃m ( t, m ∈ W n &D m ⊆ B)).Define the enumeration operators Φ n byThis gives another definition of e-reducibility:A ≤ e B ⇔ ∃n (Φ n (B) = A).In this case we say that W n determines Φ n . Call a sequence {Θ n } n∈ω of enumeration operators computable whenever there exists a computable sequence {A n } n∈ω of computably enumerable sets that determine Θ n . An arbitrary nonempty family I of e-degrees of sets of naturals is called an e-ideal whenever the following are fulfilled: 1) a ≤ e b and b ∈ I ⇒ a ∈ I; 2) a, b ∈ I ⇒ a b ∈ I. Denote by (M n ) = the set of all n-tuples of pairwise distinct elements of M ; i.e., (M n ) = = {ā ∈ M n | a i = a j for i < j}.We will assume that if M 0 and M 1 are models of distinct signatures then M 0 cannot be embedded into M 1 .

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Some Notes on Degree Spectra of the Structures

Kalimullin

2007

Lecture Notes in Computer Science

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S-Subsets of Natural Numbers

Cited by 28 publications

References 7 publications

Descriptive properties on admissible sets

Descriptive properties on admissible sets

On Σ-subsets of naturals over abelian groups

Some Notes on Degree Spectra of the Structures

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