Turing degrees of certain isomorphic images of computable relations
The author continues her investigations of the possible spectra of a relation \(R\) on a computable structure \(A\). Here the spectrum is defined as the possible degrees of images of \(R\) in computable copies of \(A\). She gives conditions for \(\text{spec}(R)\) to be the Turing degrees. These conditions give as a corollary the result independently obtained by \textit{C. J. Ash}, \textit{P. Cholak}, and \textit{J. F. Knight} [``Permitting, forcing, and copying of a given recursive relation'', Ann. Pure Appl. Log. 86, No. 3, 219-236 (1997; Zbl 0883.03029)] that if \(R\) has copies for all \(\Delta_3\) degrees, via isomorphisms of the same degrees, then the spectrum is all of the degrees. The proof also uses genericity. The author also gives an example, based on work of Jockusch, that there is a relation \(R\) whose spectrum is all the \(\Delta_2\) degrees.
scientific article; zbMATH DE number 1308069
scientific article