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PreprintCC BY 4.0arXiv

An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees (short version)

  • Jean Abou Samra
  • David A. Madore
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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. This is a short version of arXiv:2603.19946, focused on the main results.

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Cite

BibTeXRIS
BibTeX
@misc{samra2026orderreversing,
  title = {{An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees (short version)}},
  author = {Jean Abou Samra and David A. Madore},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08319},
  eprint = {2611.08319},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08319},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees (short version)
AU  - Jean Abou Samra
AU  - David A. Madore
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08319
UR  - https://arxiv.org/abs/2611.08319
AB  - 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. This is a short version of arXiv:2603.19946, focused on the main results.
ER  -