Degrees that are not degrees of categoricity
Bernard A. Anderson, Barbara F. Csima

TL;DR
This paper investigates the concept of degrees of categoricity in computable structures, constructing examples and identifying classes of degrees that are not degrees of categoricity, including certain noncomputable and hyperimmune-free degrees.
Contribution
It constructs a Sigma_2 set with a degree not of categoricity and identifies broad classes of degrees that lack this property, advancing understanding of computable structure categoricity.
Findings
Constructed a Sigma_2 set with a non-categoricity degree.
Showed that degrees of 2-generic sets relative to perfect trees are not degrees of categoricity.
Proved that all noncomputable hyperimmune-free degrees are not degrees of categoricity.
Abstract
A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a computable A such that A is x-computably categorical, and for all y, if A is y-computably categorical then y computes x. We construct a Sigma_2 set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
