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An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees

Jean Abou Samra, David Alexander Madore

Abstract

The Arthur-Nimue-Merlin degrees are a generalization of the Turing degrees introduced by Kihara as a tangible description of the partially ordered set of Lawvere-Tierney topologies on the effective topos (equivalently, subtoposes of the effective topos). They are defined in terms of a three-player game that introduces both angelic and demonic non-determinism into oracle queries. We construct an order embedding of the Turing degrees with their order reversed into the Arthur-Nimue-Merlin degrees, whose image we call the "co-Turing degrees"; we then study the order relationship of these co-Turing degrees with the (naturally embedded) Turing degrees within the Arthur-Nimue-Merlin degrees.

An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees

Abstract

The Arthur-Nimue-Merlin degrees are a generalization of the Turing degrees introduced by Kihara as a tangible description of the partially ordered set of Lawvere-Tierney topologies on the effective topos (equivalently, subtoposes of the effective topos). They are defined in terms of a three-player game that introduces both angelic and demonic non-determinism into oracle queries. We construct an order embedding of the Turing degrees with their order reversed into the Arthur-Nimue-Merlin degrees, whose image we call the "co-Turing degrees"; we then study the order relationship of these co-Turing degrees with the (naturally embedded) Turing degrees within the Arthur-Nimue-Merlin degrees.
Paper Structure (8 sections, 36 theorems, 16 equations)

This paper contains 8 sections, 36 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Arthur–Nimue–Merlin games and degrees
  3. Lattice structure on the degrees
  4. Decoding and coding degrees associated to promise problems
  5. Co-Turing degrees and their order
  6. Comparing co-Turing degrees with Turing degrees
  7. Topos-theoretic reformulation of some constructions
  8. Questions and further discussion

Key Result

Proposition 2.5

T3 reducibility $\mathrel{\preceq_{\mathrm{T3}}}$ is a preorder (i.e., reflexive and transitive) relation between functions $\mathbb{N} \to \mathcal{P}(\mathcal{P}(\mathbb{N}))$. In particular, T3 equivalence $\mathrel{\equiv_{\mathrm{T3}}}$ is an equivalence relation.

Theorems & Definitions (108)

  • Definition 2.1
  • Definition 2.3
  • Proposition 2.5
  • proof : Reference
  • Definition 2.6
  • Definition 2.8
  • proof : Solution
  • Proposition 3.1
  • proof
  • Proposition 3.3
  • ...and 98 more