The subTuring degrees

Takayuki Kihara; Keng Meng Ng

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arXiv · 2411.06043

The subTuring degrees

Takayuki Kihara·Keng Meng Ng

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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 (that is, 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 decomposed into two smaller realizability subtoposes).

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BibTeX RIS

Takayuki Kihara, Keng Meng Ng. 2024-11-09. The subTuring degrees. https://arxiv.org/abs/2411.06043

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