Degree Spectra of Structures Relative to Equivalences

Abstract

A standard way to capture the inherent complexity of the isomorphism type of a countable structure is to consider the set of all Turing degrees relative to which the given structure has a computable isomorphic copy. This set is called the degree spectrum of a structure. Similarly, to characterize the complexity of models of a theory, one may examine the set of all degrees relative to which the theory has a computable model. Such a set of degrees is called the degree spectrum of a theory. We generalize these two notions to arbitrary equivalence relations. For a structure $$ \mathcal{A} $$ and an equivalence relation E, the degree spectrum DgSp( $$ \mathcal{A} $$ , E) of $$ \mathcal{A} $$ relative to E is defined to be the set of all degrees capable of computing a structure $$ \mathcal{B} $$ that is E-equivalent to $$ \mathcal{A} $$ . Then the standard degree spectrum of $$ \mathcal{A} $$ is DgSp( $$ \mathcal{A} $$ , ≅) and the degree spectrum of the theory of $$ \mathcal{A} $$ is DgSp( $$ \mathcal{A} $$ , ≡). We consider the relations $$ {\equiv}_{\sum_n} $$ ( $$ \mathcal{A}{\equiv}_{\sum_n}\mathcal{B} $$ iff the Σn theories of $$ \mathcal{A} $$ and $$ \mathcal{B} $$ coincide) and study degree spectra with respect to $$ {\equiv}_{\sum_n} $$ .