Degrees that 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.
Keywords
Cite
@article{arxiv.1210.4220,
title = {Degrees that are not degrees of categoricity},
author = {Bernard A. Anderson and Barbara F. Csima},
journal= {arXiv preprint arXiv:1210.4220},
year = {2016}
}
Comments
10 pages. Minor revisions