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The subTuring degrees

Part of: Computability and recursion theory

Published online by Cambridge University Press:  17 December 2025

Takayuki Kihara Open the ORCID record for Takayuki Kihara [Opens in a new window]  and
Keng Meng Ng Open the ORCID record for Keng Meng Ng [Opens in a new window]
Takayuki Kihara*
Affiliation:
Nagoya University , Japan
Keng Meng Ng
Affiliation:
Nanyang Technological University , Singapore e-mail: kmng@ntu.edu.sg
*
e-mail: kihara@i.nagoya-u.ac.jp
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Abstract

In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. We show that the subTuring degrees (i.e., the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. We also show that there is a nonzero join-irreducible subTuring degree (which implies that there is a realizability subtopos of the effective topos that cannot be the meet of two larger realizability subtoposes of the effective topos).

Keywords

Turing degreerealizabilitypartial function

MSC classification

Primary: 03D30: Other degrees and reducibilities
Secondary: 03D80: Applications of computability and recursion theory

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 26
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first-named author was supported by JSPS KAKENHI (Grant Numbers 21H03392, 22K03401, and 23K28036). The second-named author was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 2 (MOE-T2EP20222-0018) and Academic Research Fund Tier 1 (RG104/24).

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