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The subTuring degrees
Logic · Submitted 9 Nov 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the subTuring degrees (that is, the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice.
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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).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. | contribution 1.00 | 0.93 | 0.06 | asserted 0.98 |
| s1 | One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. | interpretation 0.56 | 0.44 | 0.12 | asserted 0.99 |
| s2 | We show that the subTuring degrees (that is, the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. | result 1.00 | 0.95 | 0.51 | asserted 1.00 |
| s3 | 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). | result 0.99 | 0.95 | 0.15 | asserted 1.00 |
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Original 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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The notion of computability relative to a partial function used here is precisely the notion of subTuring reducibility (Definition [ref]), which is a slight modification of Sasso's definition (first formally introduced by Madore [ref] and later by Kihara [ref] ).
Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists · Background · 0.82
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