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 Algebra Logika, 2016, Volume 55, Number 6, Pages 769–799 (Mi al774)

Generalized hyperarithmetical computability over structures

A. I. Stukachevab

a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia

Abstract: We consider the class of approximation spaces generated by admissible sets and, in particular, by hereditarily finite superstructures over structures. Generalized computability on approximation spaces is conceived of as effective definability in dynamic logic. By analogy with the notion of a structure $\Sigma$-definable in an admissible set, we introduce the notion of a structure effectively definable on an approximation space. In much the same way as the $\Sigma$-reducibility relation, we can naturally define a reducibility relation on structures generating appropriate semilattices of degrees of structures (of arbitrary cardinality), as well as a jump operation. It is stated that there is a natural embedding of the semilattice of hyperdegrees of sets of natural numbers in the semilattices mentioned, which preserves the hyperjump operation. A syntactic description of structures having hyperdegree is given.

Keywords: computability theory, admissible sets, approximation spaces, constructive models, computable analysis, hyperarithmetical computability.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-05114-a Ministry of Education and Science of the Russian Federation ÍØ-6848.2016.1 Supported by RFBR (project No. 15-01-05114-a) and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1).

DOI: https://doi.org/10.17377/alglog.2016.55.606

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English version:
Algebra and Logic, 2017, 55:6, 507–526

Bibliographic databases:

UDC: 510.5
Revised: 07.11.2016

Citation: A. I. Stukachev, “Generalized hyperarithmetical computability over structures”, Algebra Logika, 55:6 (2016), 769–799; Algebra and Logic, 55:6 (2017), 507–526

Citation in format AMSBIB
\Bibitem{Stu16} \by A.~I.~Stukachev \paper Generalized hyperarithmetical computability over structures \jour Algebra Logika \yr 2016 \vol 55 \issue 6 \pages 769--799 \mathnet{http://mi.mathnet.ru/al774} \crossref{https://doi.org/10.17377/alglog.2016.55.606} \transl \jour Algebra and Logic \yr 2017 \vol 55 \issue 6 \pages 507--526 \crossref{https://doi.org/10.1007/s10469-017-9421-1} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000396462200006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85014996007} 

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This publication is cited in the following articles:
1. A. I. Stukachev, “Processes and structures on approximation spaces”, Algebra and Logic, 56:1 (2017), 63–74
2. A. I. Stukachev, “Intervalnye rasshireniya poryadkov i temporalnye approksimatsionnye prostranstva”, Sib. matem. zhurn., 62:4 (2021), 894–910
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