Primitive recursive categoricity spectra of functional structures

TL;DR

This paper investigates the degrees of categoricity for punctual structures, establishing their relation to Turing degrees and constructing specific examples to illustrate differences in categoricity notions.

Contribution

It introduces a new notion of degree of categoricity for punctual structures, compares it with existing notions, and constructs examples demonstrating their divergence.

Findings

Non-$ riangle_{1}^{0}$-categorical injection structures have coinciding notions.

Existence of $ riangle_{1}^{0}$-categorical injection structures where notions differ.

In every non-zero c.e. Turing degree, PR-degrees that are low for punctual isomorphism and degrees of punctual categoricity exist.

Abstract

For the notion of degree of categoricity, we study an analogous notion for punctual structures. We show that such notions coincide for non-$\Delta_{1}^{0}$-categorical injection structures, and construct an example of a $\Delta_{1}^{0}$-categorical injection structure for which these notions differ. Additionally, we also show that in every non-zero c.e.~Turing degree, there exists a PR-degree that is low for punctual isomorphism (to be defined), and also a PR-degree that is a degree of punctual categoricity.