Reductions between types of numberings

Herbert, Ian; Jain, Sanjay; Lempp, Steffen; Mustafa, Manat; Stephan, Frank

doi:10.1016/j.apal.2019.102716

Cited by 12 publications

(8 citation statements)

References 27 publications

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“…Following the approach of [24], we here discuss reductions between types of numberings. Our first reduction Γ (§ 3.1) transforms a

\Sigma ^0_2

‐computable family into an l.m.…”

Section: Reductions For Normalς20$\sigma ^0_2$‐computable Numberingsmentioning

confidence: 99%

“…[20]. For more results on computable numberings, the reader is referred to the seminal monograph [17] and the papers [1,2,18,24].…”

Section: Introductionmentioning

confidence: 99%

“…During the last two decades, this approach has become a fruitful line of research with plenty of highly nontrivial results. The research direction is focused on uniform computations in various recursion-theoretic hierarchies: the arithmetical hierarchy (cf., e.g., [1,2,5]), the analytical hierarchy [7,10], the Ershov hierarchy [6,22,24], etc.…”

Section: Introductionmentioning

confidence: 99%

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Rogers semilattices of limitwise monotonic numberings

Bazhenov

Mustafa

Tleuliyeva

2022

Mathematical Logic Qtrly

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Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S⊂Pfalse(ωfalse)$S\subset P(\omega )$ is limitwise monotonic (l.m.) if every set νfalse(kfalse)$\nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice Rlm(S)$R_{lm}(S)$. The semilattices Rlm(S)$R_{lm}(S)$ exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of normalΣ20$\Sigma ^0_2$‐computable families. We show that every Rogers semilattice of a normalΣ20$\Sigma ^0_2$‐computable family is isomorphic to some semilattice Rlm(S)$R_{lm}(S)$. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices Rlm(S)$R_{lm}(S)$. In particular, there is an l.m. family S such that Rlm(S)$R_{lm}(S)$ is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset Rlm(S)$R_{lm}(S)$ is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all normalΣ20$\Sigma ^0_2$‐computable numberings. We prove that inside this class, the index set of l.m. numberings is normalΣ40$\Sigma ^0_4$‐complete.

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“…Following the approach of [24], we here discuss reductions between types of numberings. Our first reduction Γ (§ 3.1) transforms a

\Sigma ^0_2

‐computable family into an l.m.…”

Section: Reductions For Normalς20$\sigma ^0_2$‐computable Numberingsmentioning

confidence: 99%

“…[20]. For more results on computable numberings, the reader is referred to the seminal monograph [17] and the papers [1,2,18,24].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rogers semilattices of limitwise monotonic numberings

Bazhenov

Mustafa

Tleuliyeva

2022

Mathematical Logic Qtrly

Self Cite

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“…sets. The result of Herbert, Jain, Lempp, Mustafa, and Stephan (Theorem 2 in [23]) implies that every Rogers Σ −1 n -semilattice is also a Rogers Σ −1 n+1 -semilattice.…”

Section: Introductionmentioning

confidence: 96%

Isomorphism types of Rogers semilattices in the analytical hierarchy

Bazhenov¹,

Ospichev²,

Yamaleev³

2019

Preprint

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A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families S ⊂ P (ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers m = n, any non-trivial Rogers semilattice of a Π 1 m -computable family cannot be isomorphic to a Rogers semilattice of a Π 1 n -computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.

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“…For a more general setting, the problem can be formulated as follows: What are the structural properties of a family of sets in the Ershov hierarchy that guarantee that the Rogers semilattice is one-element? In view of the properties of the F. Stephan operator [13], it suffices to seek a solution to this question among families of sets at two lower levels in the Ershov hierarchy.…”

mentioning

confidence: 99%

One-Element Rogers Semilattices in the Ershov Hierarchy

et al. 2021

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The cardinality of the Rogers semilattice of a computable family of sets is its most natural invariant. The cardinality problem for semilattices of computable numberings, which was raised by Ershov [1] for families of computably enumerable (c.e.) sets, has long been canonical. It arises every time when more and more classes of families of objects become involved in the field of research.A fundamental answer to the question about the cardinality of Rogers semilattices for the classical case due to A. B. Khutoretskii [2] reads as follows: the semilattice of a computable family of c.e. sets is either one-element or infinite. An extensive bibliography on this subject is contained in [3][4][5][6]. All known results confirm Khutoretskii's dichotomy: the semilattice contains either at most one or infinitely many elements.And what is the reason for the above-mentioned dichotomy? Research in this direction was initiated by Mal'tsev [7] and Ershov [1], who tried to find a topological characterization of families with a one-element semilattice of computable numberings, and was continued in [8][9][10][11]. A complete * Supported by KN MON RK, grant No. AP08856834.

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Reductions between types of numberings

Cited by 12 publications

References 27 publications

Rogers semilattices of limitwise monotonic numberings

Rogers semilattices of limitwise monotonic numberings

Isomorphism types of Rogers semilattices in the analytical hierarchy

One-Element Rogers Semilattices in the Ershov Hierarchy

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