Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913258962681856 |
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| author | Kihara, Takayuki |
| author_facet | Kihara, Takayuki |
| contents | We present three different perspectives of oracle. First, an oracle is a blackbox; second, an oracle is a tool to change the way we access mathematical objects; and third, an oracle is a factor that causes a change in truth values. Formally, the second perspective advocates that an oracle is an endofunctor on the category of coded sets (preserving underlying sets) -- we associate it with a universal closure operator. The third perspective advocates that an oracle is an operation on the object of truth values -- we associate it with a Lawvere-Tierney topology. These three perspectives create a link between the three fields, computability theory, synthetic descriptive set theory, and effective topos theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_00188 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists Kihara, Takayuki Logic Logic in Computer Science We present three different perspectives of oracle. First, an oracle is a blackbox; second, an oracle is a tool to change the way we access mathematical objects; and third, an oracle is a factor that causes a change in truth values. Formally, the second perspective advocates that an oracle is an endofunctor on the category of coded sets (preserving underlying sets) -- we associate it with a universal closure operator. The third perspective advocates that an oracle is an operation on the object of truth values -- we associate it with a Lawvere-Tierney topology. These three perspectives create a link between the three fields, computability theory, synthetic descriptive set theory, and effective topos theory. |
| title | Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2202.00188 |