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DECOMPOSITION OF GÖDEL NUMBERINGS INTO MINIMAL NUMBERINGS

Part of: Computability and recursion theory

Published online by Cambridge University Press:  24 November 2025

MARAT FAIZRAHMANOV Open the ORCID record for MARAT FAIZRAHMANOV [Opens in a new window]
MARAT FAIZRAHMANOV*
Affiliation:
KAZAN FEDERAL UNIVERSITY RUSSIA
*
E-mail: marat.faizrahmanov@gmail.com
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Abstract

This article studies the question of which classes of numberings can be generated by the direct sums of computable uniformly minimal sequences of numberings (in particular, Friedberg and positive numberings). For the class of Friedberg numberings, this question was initiated by Britta Schinzel in her 1982 paper. In this article, we show that the class of Gödel numberings is generated by the direct sums of computable uniformly positive sequences of universal numberings and that there exists a conull class of oracles computing sequences of Friedberg numberings with programmable direct sums. We further show that all computable numberings of a fairly wide class of families of total recursive functions (containing, for example, the family of all primitive recursion functions) are generated by the direct sums of computable sequences of their incomparable Friedberg numberings. On the other hand we prove that no family of partial recursive functions has a computable sequence of Friedberg numberings whose direct sum is acceptable.

Keywords

computable numberingGödel numberingminimal numberingpositive numberingFriedberg numbering

MSC classification

Primary: 03D45: Theory of numerations, effectively presented structures 03D20: Recursive functions and relations, subrecursive hierarchies
Secondary: 03D30: Other degrees and reducibilities

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 19
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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