A Relation on ${(\omega, <)}$ of Intermediate Degree Spectrum on a Cone

TL;DR

This paper constructs a natural relation on the ordered set of natural numbers that has an intermediate degree spectrum, filling a gap in understanding the possible Turing degree spectra of relations.

Contribution

It provides the first natural example of a relation with an intermediate degree spectrum on a cone, using structural properties rather than diagonalization.

Findings

Constructed a natural relation with intermediate degree spectrum.

Showed the relation's spectrum is due to structural reasons.

Fills a gap in the classification of degree spectra of relations.

Abstract

We examine the degree spectra of relations on ${(\omega, <)}$. Given an additional relation $R$ on ${(\omega,<)}$, such as the successor relation, the degree spectrum of $R$ is the set of Turing degrees of $R$ in computable copies of ${(\omega,<)}$. It is known that all degree spectra of relations on ${(\omega,<)}$ fall into one of four categories: the computable degree, all of the c.e. degrees, all of the $\Delta^0_2$ degrees, or intermediate between the c.e. degrees and the $\Delta^0_2$ degrees. Examples of the first three degree spectra are easy to construct and well-known, but until recently it was open whether there is a relation with intermediate degree spectrum on a cone. Bazhenov, Kaloci\'{n}ski, and Wroclawski constructed an example of an intermediate degree spectrum, but their example is unnatural in the sense that it is constructed by diagonalization and thus not canonical, that is, which relation you obtain from their construction depends on which G\"odel encoding (and hence order of enumeration) of the partial computable functions / programs you choose. In this paper, we use the ''on-a-cone'' paradigm to restrict our attention to "natural" relations $R$. Our main result is a construction of a natural relation on ${(\omega,<)}$ which has intermediate degree spectrum. This relation has intermediate degree spectrum because of structural reasons.