Turing degrees of hypersimple relations on computable structures Original Research Article

Let View the MathML source be an infinite computable structure, and let R be an additional computable relation on its domain A. The syntactic notion of formal hypersimplicity of R on View the MathML source, first introduced and studied by Hird, is analogous to the computability-theoretic notion of hypersimplicity of R on A, given the definability of certain effective sequences of relations on A. Assuming that R is formally hypersimple on View the MathML source, we give general sufficient conditions for the existence of a computable isomorphic copy of View the MathML source on whose domain the image of R is hypersimple and of arbitrary nonzero computably enumerable Turing degree.