Rethinking the notion of oracle

Kihara, Taro

doi:10.48550/arxiv.2202.00188

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…Remark 3.7. After completing this article, the author succeeded in giving a crystalclear explanation of the results in this section (the correspondence between Lawvere-Tierney topologies and oracles); see [24].…”

Section: Theorem 31 the Poset Reflections Of The Following Preorders ...mentioning

confidence: 99%

Untitled

2023

Trans. Amer. Math. Soc. Ser. B

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In this article, we study the lattice of Lawvere-Tierney topologies on Hyland's effective topos. For this purpose, we introduce a new computabilitytheoretic reducibility notion, which is a common extension of the notions of Turing reducibility and generalized Weihrauch reducibility. Based on the work by Lee and van Oosten [Ann. Pure Appl. Logic 164 (2013), pp. 866-883], we utilize this reducibility notion for providing a concrete description of the lattice of the Lawvere-Tierney topologies on the effective topos. As an application, we solve several open problems proposed by Lee and van Oosten. For instance, we show that there exists no minimal Lawvere-Tierney topology which is strictly above the identity topology on the effective topos.

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Section: Theorem 31 the Poset Reflections Of The Following Preorders ...mentioning

confidence: 99%

Untitled

2023

Trans. Amer. Math. Soc. Ser. B

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Section: Theorem 31 the Poset Reflections Of The Following Preorders ...mentioning

confidence: 99%

Lawvere-Tierney topologies for computability theorists

Kihara

2023

Trans. Amer. Math. Soc. Ser. B

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In this article, we study the lattice of Lawvere-Tierney topologies on Hyland’s effective topos. For this purpose, we introduce a new computability-theoretic reducibility notion, which is a common extension of the notions of Turing reducibility and generalized Weihrauch reducibility. Based on the work by Lee and van Oosten [Ann. Pure Appl. Logic 164 (2013), pp. 866-883], we utilize this reducibility notion for providing a concrete description of the lattice of the Lawvere-Tierney topologies on the effective topos. As an application, we solve several open problems proposed by Lee and van Oosten. For instance, we show that there exists no minimal Lawvere-Tierney topology which is strictly above the identity topology on the effective topos.

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Rethinking the notion of oracle

Cited by 2 publications

References 19 publications

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Lawvere-Tierney topologies for computability theorists

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