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

TL;DR

This paper constructs an order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees, revealing new structural relationships and introducing the concept of co-Turing degrees within this framework.

Contribution

It provides the first known order embedding of Turing degrees into Arthur-Nimue-Merlin degrees with reversed order, expanding understanding of their structure.

Findings

Established an order-reversing embedding of Turing degrees.

Defined and analyzed the co-Turing degrees within Arthur-Nimue-Merlin degrees.

Explored the relationship between co-Turing and Turing 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.