An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees
This work addresses a foundational problem in computability theory for researchers studying degree structures and topos theory, representing an incremental advancement in understanding these generalized degrees.
The paper tackles the problem of embedding Turing degrees into Arthur-Nimue-Merlin degrees by constructing an order-reversing embedding, resulting in the definition of 'co-Turing degrees' and an analysis of their order relationships within the broader structure.
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.