INFINITE FAMILIES OF TOTAL FUNCTIONS WITH PRINCIPAL NUMBERINGS

Alibek Issakhov, F. Rakymzhankyzy, Uldana Ostemirova

Herald of Kazakh-British technical university, 2021

Full text Open access: Gold Metadata (CSL-JSON)

DOI 10.55452/1998-6688-2021-18-2-53-58

Data confidence 70 medium

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Abstract

It was known that any non-single-element (in particular, any infinite) family of total functions with an oracle , such that , does not have -computable principal numbering; later it was proved that any finite family of total functions with a hyperimmune-free oracle always has an -computable principal numbering. The unresolved question was whether there exists an infinite family of total functions with a hyperimmune-free oracle that has an -computable principal numbering. The paper gives a positive answer to this question: it is proved that there exists an infinite -computable family of total functions, where the Turing degree of the set is hyperimmune-free, such that has an -computable principal numbering.