TY - GEN

T1 - Semilattices of Punctual Numberings

AU - Bazhenov, Nikolay

AU - Mustafa, Manat

AU - Ospichev, Sergei

N1 - Funding Information: The work was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342. The first author was partially supported by the grant of the President of the Russian Federation (No. MK-1214.2019.1). The third author was partially supported by the program of fundamental scientific researches of the SB RAS No. I.1.1, project No. 0314-2019-0002. Publisher Copyright: © 2020, Springer Nature Switzerland AG. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - The theory of numberings studies uniform computations for classes of mathematical objects. A large body of literature is devoted to investigations of computable numberings, i.e. uniform enumerations for families of computably enumerable sets, and the reducibility among these numberings. This reducibility, induced by Turing computable functions, aims to classify the algorithmic complexity of numberings. The paper is inspired by the recent advances in the area of punctual algebraic structures. We recast the classical studies of numberings in the punctual setting—we study punctual numberings, i.e. uniform computations for families of primitive recursive functions. The reducibility between punctual numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We prove that any infinite Rogers pr-semilattice is dense and does not have minimal elements. Furthermore, we give an example of infinite Rogers pr-semilattice, which is a lattice. These results exhibit interesting phenomena, which do not occur in the classical case of computable numberings and their semilattices.

AB - The theory of numberings studies uniform computations for classes of mathematical objects. A large body of literature is devoted to investigations of computable numberings, i.e. uniform enumerations for families of computably enumerable sets, and the reducibility among these numberings. This reducibility, induced by Turing computable functions, aims to classify the algorithmic complexity of numberings. The paper is inspired by the recent advances in the area of punctual algebraic structures. We recast the classical studies of numberings in the punctual setting—we study punctual numberings, i.e. uniform computations for families of primitive recursive functions. The reducibility between punctual numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We prove that any infinite Rogers pr-semilattice is dense and does not have minimal elements. Furthermore, we give an example of infinite Rogers pr-semilattice, which is a lattice. These results exhibit interesting phenomena, which do not occur in the classical case of computable numberings and their semilattices.

KW - Friedberg numbering

KW - Numbering

KW - Online computation

KW - Primitive recursion

KW - Punctual structure

KW - Rogers semilattice

KW - Upper semilattice

UR - http://www.scopus.com/inward/record.url?scp=85093854226&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-59267-7_1

DO - 10.1007/978-3-030-59267-7_1

M3 - Conference contribution

AN - SCOPUS:85093854226

SN - 9783030592660

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 12

BT - Theory and Applications of Models of Computation - 16th International Conference, TAMC 2020, Proceedings

A2 - Chen, Jianer

A2 - Feng, Qilong

A2 - Xu, Jinhui

PB - Springer Science and Business Media Deutschland GmbH

T2 - 16th Annual Conference on Theory and Applications of Models of Computation, TAMC 2020

Y2 - 18 October 2020 through 20 October 2020

ER -