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 Sibirsk. Mat. Zh., 2018, Volume 59, Number 4, Pages 823–833 (Mi smj3012)

Positive presentations of families relative to $e$-oracles

I. Sh. Kalimullina, V. G. Puzarenkob, M. Kh. Faizrahmanova

a Kazan (Volga Region) Federal University, Kazan, Russia
b Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: We introduce the notion of $A$-numbering which generalizes the classical notion of numbering. All main attributes of classical numberings are carried over to the objects considered here. The problem is investigated of the existence of positive and decidable computable $A$-numberings for the natural families of sets $e$-reducible to a fixed set. We prove that, for every computable $A$-family containing an inclusion-greatest set, there also exists a positive computable $A$-numbering. Furthermore, for certain families we construct a decidable (and even single-valued) computable total $A$-numbering when $A$ is a low set; we also consider a relativization containing all cases of total sets (this in fact corresponds to computability with a usual oracle).

Keywords: enumeration, decidable numbering, positive numbering, computable numbering, computable set, computably enumerable set, $e$-reducibility.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.451.2016/1.41.1515.2017/4.6 Russian Foundation for Basic Research 18-01-00624 I. Sh. Kalimullin was supported by the subsidy of the Government Task for Kazan (Volga Region) Federal University (Grant 1.451.2016/1.4). V. G. Puzarenko was supported by the Russian Foundation for Basic Research (Grant 18-01-00624). M. Kh. Faizrahmanov was supported by the subsidy of the Government Task for Kazan (Volga Region) Federal University (Grant 1.1515.2017/4.6).

DOI: https://doi.org/10.17377/smzh.2018.59.407

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English version:
Siberian Mathematical Journal, 2018, 59:4, 648–656

Bibliographic databases:

UDC: 510.5
MSC: 35R30

Citation: I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Faizrahmanov, “Positive presentations of families relative to $e$-oracles”, Sibirsk. Mat. Zh., 59:4 (2018), 823–833; Siberian Math. J., 59:4 (2018), 648–656

Citation in format AMSBIB
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This publication is cited in the following articles:
1. I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Faizrahmanov, “Partial decidable presentations in hyperarithmetic”, Siberian Math. J., 60:3 (2019), 464–471
2. I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Faizrakhmanov, “O pozitivnykh i odnoznachnykh vychislimykh numeratsiyakh v giperarifmetike”, Algebra i logika, 59:1 (2020), 66–83
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