Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Takayuki Kihara

TL;DR

Three different perspectives of oracle are presented, which create a link between the three fields, computability theory, synthetic descriptive set theory, and effective 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.

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

TL;DR

Three different perspectives of oracle are presented, which create a link between the three fields, computability theory, synthetic descriptive set theory, and effective 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.
Paper Structure (35 sections, 42 theorems, 45 equations, 2 tables)

This paper contains 35 sections, 42 theorems, 45 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Three perspectives of oracle
  3. Computability-Theoretic Background
  4. Oracle and Turing reducibility
  5. Mass problems
  6. Search problems
  7. Reverse mathematics and logic
  8. Turing-like reducibility for search problems
  9. Related notions in other areas
  10. Topos theory
  11. Degree theory encounters with topos theory
  12. Overview
  13. Preliminaries
  14. Abstract characterization of universal oracle computation
  15. Universal machine
  16. ...and 20 more sections

Key Result

Proposition 2.11

The order-preserving map ${\tt Weih}\colon\mathcal{MR}ed\to\mathcal{MR}ed^{\rm ct}$ is left adjoint to the inclusion $i\colon\mathcal{MR}ed^{\rm ct}\rightarrowtail\mathcal{MR}ed$. In other words, for any $f,g\colon\!\!\!\subseteq\raisebox{0pt}{$\underset{\widetilde{}}{\mathbf{N}}$}\rightrightarrows\

Theorems & Definitions (167)

  • Definition 2.1: de Brecht dB14
  • Example 2.2: when $\raisebox{0pt}{$\underset{\widetilde{}}{\mathbf{N}}$}=\mathbb{N}^\mathbb{N}$
  • Remark
  • Remark
  • Definition 2.3
  • Example 2.4: Medvedev oracle
  • Example 2.5: Weihrauch oracle
  • Remark
  • Definition 2.6
  • Remark : Lifschitz realizability
  • ...and 157 more