Turing degrees measure computational difficulty: A is harder than B if computing A requires an oracle for B. The ordering goes upward — more complex, more powerful, more capable. The hierarchy is monotone.
Researchers construct an order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees — a richer computational hierarchy introduced by Kihara as a concrete description of Lawvere-Tierney topologies on the effective topos. The image of this embedding defines “co-Turing degrees,” where the ordering runs backward: what was easy becomes hard, what was hard becomes easy.
The reversal isn't a trick of notation. Arthur-Nimue-Merlin degrees arise from a three-player game combining angelic non-determinism (Arthur), demonic non-determinism (Nimue), and verification (Merlin). In this game-theoretic framework, the oracle queries involve both cooperative and adversarial access — you can ask questions, but the answers come through a channel that includes both help and obstruction. The interplay between these forces creates a degree structure rich enough to embed the Turing degrees in reverse.
The mathematical content: Lawvere-Tierney topologies classify the logical subtoposes of the effective topos — essentially, the different “worlds” of computable mathematics. Kihara showed these topologies correspond to Arthur-Nimue-Merlin degrees, making the degree structure a concrete representation of something deeply abstract. The order-reversal embedding means that within this space, there exists a mirror image of classical computability where complexity runs backward.
The structural lesson: a degree structure is an ordering, and orderings can embed in unexpected ways into richer structures. The Turing degrees seem canonical — the natural hierarchy of computational difficulty. But they're one projection of a higher-dimensional landscape where “harder” and “easier” can reverse depending on which axis you project along. The reversal is real. The apparent naturalness of the original ordering was parochial.