Rethinking the notion of oracle: A prequel to Lawvere-Tierney topologies for computability theorists

TL;DR

This paper explores three conceptual perspectives of oracles—blackboxes, access tools, and truth changers—and formalizes them using category theory, linking computability, descriptive set theory, and topos theory.

Contribution

It introduces formal categorical frameworks for oracles, connecting different fields and providing a unified understanding of their roles in computation and logic.

Findings

Formalizes oracle as an endofunctor on coded sets

Associates oracle with a universal closure operator

Links oracle to Lawvere-Tierney topologies in topos theory

Abstract

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.