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Primitive recursive categoricity spectra of functional structures

Nikolay Bazhenov, Heer Tern Koh, Keng Meng Ng

Abstract

For the notion of degree of categoricity, we study an analogous notion for punctual structures. We show that such notions coincide for non-$Δ_{1}^{0}$-categorical injection structures, and construct an example of a $Δ_{1}^{0}$-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.

Primitive recursive categoricity spectra of functional structures

Abstract

For the notion of degree of categoricity, we study an analogous notion for punctual structures. We show that such notions coincide for non--categorical injection structures, and construct an example of a -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.
Paper Structure (5 sections, 7 theorems, 3 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Injection Structures
  4. A pathological injection structure
  5. Degrees of punctual categoricity

Key Result

Theorem 2.1

Let $A$ be a computable injection structure with at least one infinite orbit. Then $A$ has a punctual presentation.

Figures (2)

  • Figure 1: The $e^{th}$ component of $B$. The function arrows for $R$ and $P$ are not shown here. The solid arrows represent the function $S$, and the dashed arrows represent $C$.
  • Figure 2: Switching in the $e^{th}$ component. The arrows represent the function $P$.

Theorems & Definitions (17)

  • Definition 1.1: km21
  • Definition 1.2: bk21
  • Definition 1.3: bk21
  • Theorem 2.1: folklore
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 7 more