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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).

References

Ackerman, N. L., Freer, C. E., and Lubarsky, R. S., An introduction to feedback Turing computability . J. Logic Comput. 30(2020), no. 1, 27–60.Google Scholar
Beeson, M., Goodman’s theorem and beyond . Pacific J. Math. 84(1979), no. 1, 1–16.Google Scholar
Chong, C. T. and Liang, Y., Recursion theory: Computational aspects of definability, volume 8 of De Gruyter Series in Logic and its Applications, De Gruyter, Berlin, 2015. With an interview with Gerald E. Sacks.Google Scholar
Cooper, S. B., Computability theory, Chapman & Hall/CRC, Boca Raton, FL, 2004.Google Scholar
Dzhafarov, D. D., Joins in the strong Weihrauch degrees . Math. Res. Lett. 26(2019), no. 3, 749–767.Google Scholar
Faber, E. and van Oosten, J., More on geometric morphisms between realizability toposes . Theory Appl. Categ. 29(2014), 874–895.Google Scholar
Goodman, N. D., Relativized realizability in intuitionistic arithmetic of all finite types . J. Symbolic Logic 43(1978), no. 1, 23–44.Google Scholar
Hyland, J. M. E., The effective topos . In: A. S. Troelstra and D. van Dalen (eds.), The L.E.J. Brouwer centenary symposium (Noordwijkerhout, 1981), volume 110 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1982, pp. 165–216.Google Scholar
Kihara, T., Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists. Preprint, 2022. arXiv:2202.00188.Google Scholar
Kihara, T., Lawvere-Tierney topologies for computability theorists . Trans. Amer. Math. Soc. Ser. B 10(2023), 48–85.Google Scholar
Kihara, T. and Ng, K. M., Church’s thesis in subtoposes of the effective topos. In preparation, 2024.Google Scholar
Madore, D., Various notions of Turing reduction for partial functions. 2012. https://mathoverflow.net/questions/112617/various-notions-of-turing-reduction-for-partial-functions Google Scholar
Odifreddi, P., Classical recursion theory: The theory of functions and sets of natural numbers, volume 125 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1989. With a foreword by G. E. Sacks.Google Scholar
Sasso, L. P. Jr., A survey of partial degree . J. Symbolic Logic 40(1975), 130–140.Google Scholar
Soskova, M. I., The theory of the enumeration degrees, definability, and automorphisms . In: A. Rezuş (ed.), Contemporary logic and computing, volume 1 of Landscape Log, College Publications, London, 2020, pp. 706–730.Google Scholar
van den Berg, B. and van Slooten, L., Arithmetical conservation results . Indag. Math. (N.S.) 29(2018), no. 1, 260–275.Google Scholar
van Oosten, J., A semantical proof of de Jongh’s theorem . Arch. Math. Logic 31(1991), no. 2, 105–114.Google Scholar
van Oosten, J., A combinatory algebra for sequential functionals of finite type . In: S. B. Cooper and J. K. Truss (eds.), Models and computability (Leeds, 1997), volume 259 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999, pp. 389–405.Google Scholar
van Oosten, J., A general form of relative recursion . Notre Dame J. Formal Logic 47(2006), no. 3, 311–318.Google Scholar
van Oosten, J., Realizability: An introduction to its categorical side, volume 152 of Studies in Logic and the Foundations of Mathematics, Elsevier B. V., Amsterdam, 2008.Google Scholar