Extensions of categoricity relative to a degree

TL;DR

This paper investigates how the property of computable categoricity varies when relativized to different degrees, demonstrating that structures can change categoricity status depending on the degree considered.

Contribution

It extends previous machinery to show that structures can be non-categorically computable but become categorical relative to specific degrees, including non-c.e. degrees.

Findings

Constructed a structure not computably categorical but categorical relative to a 1-generic degree.

Showed that other classes of structures also exhibit degree-dependent categoricity behavior.

Abstract

In this paper, we apply the machinery developed in arXiv:2401.06641(2) to study the behavior of computable categoricity relativized to non-c.e. degrees. In particular, we show that we can build a computable structure which is not computably categorical but is computably categorical relative to a $1$-generic degree. Additionally, we show that other classes of structures besides directed graphs admit a computable example which can change its computable categorical behavior relative to different degrees.