Rogers semilattices of punctual numberings

The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducib...

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Bibliographic Details
Published in:Mathematical structures in computer science Vol. 32; no. 2; pp. 164 - 188
Main Authors: Bazhenov, Nikolay, Mustafa, Manat, Ospichev, Sergei
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.02.2022
Subjects:
Algorithms
Combinatorial analysis
Recursive functions
Special Issue: Theory and Applications of Models of Computation (TAMC 2020)
primitive recursion
decidability
Friedberg numbering
online computation
Theory of numberings
punctual structure
upper semilattice
Rogers semilattice
ISSN:0960-1295, 1469-8072
Online Access:Get full text
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Summary:The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $\Sigma_1$ -fragment of the theory Th(R) is decidable.
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ISSN:0960-1295
1469-8072
DOI:10.1017/S0960129522000093