Degree Spectra of Relations on a Cone

TL;DR

This paper studies the degree spectra of relations on structures with an emphasis on cones, revealing minimal spectra, differences in complexity levels, and answering open questions about the nature of these spectra in computability theory.

Contribution

It introduces the concept of degree spectra on a cone, analyzes their properties, and provides new results on the complexity of spectra for various degrees and structures.

Findings

Existence of a minimal non-trivial degree spectrum of c.e. degrees on a cone.

Counterexample showing degree spectra on a cone can be incomparable in complexity.

Positive answer to a question about degree spectra containing all 2-CEA degrees.

Abstract

Let $\mathcal{A}$ be a mathematical structure with an additional relation $R$. We are interested in the degree spectrum of $R$, either among computable copies of $\mathcal{A}$ when $(\mathcal{A},R)$ is a "natural" structure, or (to make this rigorous) among copies of $(\mathcal{A},R)$ computable in a large degree \textbf{d}. We introduce the partial order of degree spectra \textit{on a cone} and begin the study of these objects. Using a result of Harizanov---that, assuming an effectiveness condition on $\mathcal{A}$ and $R$, if $R$ is not intrinsically computable, then its degree spectrum contains all c.e.\ degrees---we see that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e.\ degrees. We show that this does not generalize to d.c.e.\ degrees by giving an example of two incomparable degree spectra on a cone. We also give a partial answer to a question of Ash and Knight: they asked whether (subject to some effectiveness conditions) a relation which is not intrinsically $\Delta^0_\alpha$ must have a degree spectrum which contains all of the $\alpha$-CEA degrees. We give a positive answer to this question for $\alpha = 2$ by showing that any degree spectrum on a cone which strictly contains the $\Delta^0_2$ degrees must contain all of the 2-CEA degrees. We also investigate the particular case of degree spectra on the structure $(\omega,<)$. This work represents the beginning of an investigation of the degree spectra of "natural" structures, and we leave many open questions to be answered.