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Extensions of categoricity relative to a degree

Java Darleen Villano

TL;DR

The paper addresses how computable categoricity behaves under relativization to non-c.e. degrees by constructing a computable directed graph $ a$ that is not computably categorical but becomes computably categorical relative to a $1$-generic degree $G$. The method employs a priority-tree construction with three families of strategies—$R_j$ for building $G$, $P_e$ for diagonalizing against computable isomorphisms, and $S_i$ for ensuring $G$-computable embeddings when $ a$ matches a $G$-computable copy $\M_i^G$—together with homogenization to manage changes in $G$. The main result demonstrates the existence of a computable structure whose categoricity can change when viewed from different degrees, and the paper extends these ideas to other structure classes via coding techniques, while noting limitations in certain well-studied classes like linear orders and Boolean algebras. This advances understanding of degrees of categoricity and shows that 1-generic degrees can realize nonmonotone relativized categoricity phenomena, with broader implications for the study of relative computability and embeddings in computable structure theory.

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.

Extensions of categoricity relative to a degree

TL;DR

The paper addresses how computable categoricity behaves under relativization to non-c.e. degrees by constructing a computable directed graph that is not computably categorical but becomes computably categorical relative to a -generic degree . The method employs a priority-tree construction with three families of strategies— for building , for diagonalizing against computable isomorphisms, and for ensuring -computable embeddings when matches a -computable copy —together with homogenization to manage changes in . The main result demonstrates the existence of a computable structure whose categoricity can change when viewed from different degrees, and the paper extends these ideas to other structure classes via coding techniques, while noting limitations in certain well-studied classes like linear orders and Boolean algebras. This advances understanding of degrees of categoricity and shows that 1-generic degrees can realize nonmonotone relativized categoricity phenomena, with broader implications for the study of relative computability and embeddings in computable structure theory.

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 -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.

Paper Structure

This paper contains 18 sections, 34 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Informal strategies for Theorem \ref{['thm: categoricity relative to a 1-generic']}
  3. Building a $1$-generic $G$
  4. Making $\mathcal{A}$ not computably categorical
  5. Being computably categorical relative to $G$
  6. An interaction caused by genericity
  7. Proof of Theorem \ref{['thm: categoricity relative to a 1-generic']}
  8. Requirements
  9. Construction
  10. $R_j$-strategies and outcomes
  11. $P_e$-strategies and outcomes
  12. $S_i$-strategies and outcomes
  13. Verification
  14. Lemmas for each strategy
  15. Lemmas on interactions between multiple strategies
  16. ...and 3 more sections

Key Result

Theorem 1.1

There is a computable structure $\mathcal{A}$ and c.e. degrees $0=Y_0<_{\text{T}} X_0<_{\text{T}} Y_1<_{\text{T}} X_1<_{\text{T}}\dots$ such that

Theorems & Definitions (61)

  • Theorem 1.1: Downey, Harrison-Trainor, Melnikov MR4291596
  • Theorem 1.2: villano2024computable
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Anderson, Csima AC2016
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8: Franklin, Solomon franklinsolomon2014
  • Theorem 1.9
  • Definition 2.1
  • ...and 51 more