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On Sets That Encode Themselves

Taeyoung Em

TL;DR

This work investigates self-encoding sets by mapping intricate relationships among introenumerable, introreducible, majorenumerable, majorreducible, regressive, and retraceable notions. It develops forcing methods to patch uniform witnesses into nonuniform objects, thereby constructing novel introenumerable sets and showing that uniformity can be reversed to yield nontrivial self-encoding behavior. The paper also clarifies a comprehensive separation landscape, proving several notions are distinct and providing tools to build sets with prescribed combinatorial and computability properties. These results deepen the understanding of how partial information can recover original sets and offer new avenues for constructing and separating self-encoding families in computability theory.

Abstract

Given partial information about a set, we are interested in fully recovering the original set from what is given. If a set encodes itself robustly, any partial information about the set suffices to fully recover the information about the original set. Jockusch defined a set $A$ to be introenumerable if each infinite subset of $A$ can enumerate $A$, and this is an example of a set which encodes itself. There are several other notions capturing self-encoding differently. We say $A$ is uniformly introenumerable if each infinite subset of $A$ can uniformly enumerate $A$, whereas $A$ is introreducible if each infinite subset of $A$ can compute $A$. We investigate properties of various notions of self-encoding and prove new results on their interactions. Greenberg, Harrison-Trainor, Patey, and Turetsky showed that we can always find some uniformity from an introenumerable set. We show that this can be reversed: we can construct an introenumerable set by patching up uniformity. This gives a rise to a new method of constructing a nontrivial introenumerable or introreducible set.

On Sets That Encode Themselves

TL;DR

This work investigates self-encoding sets by mapping intricate relationships among introenumerable, introreducible, majorenumerable, majorreducible, regressive, and retraceable notions. It develops forcing methods to patch uniform witnesses into nonuniform objects, thereby constructing novel introenumerable sets and showing that uniformity can be reversed to yield nontrivial self-encoding behavior. The paper also clarifies a comprehensive separation landscape, proving several notions are distinct and providing tools to build sets with prescribed combinatorial and computability properties. These results deepen the understanding of how partial information can recover original sets and offer new avenues for constructing and separating self-encoding families in computability theory.

Abstract

Given partial information about a set, we are interested in fully recovering the original set from what is given. If a set encodes itself robustly, any partial information about the set suffices to fully recover the information about the original set. Jockusch defined a set to be introenumerable if each infinite subset of can enumerate , and this is an example of a set which encodes itself. There are several other notions capturing self-encoding differently. We say is uniformly introenumerable if each infinite subset of can uniformly enumerate , whereas is introreducible if each infinite subset of can compute . We investigate properties of various notions of self-encoding and prove new results on their interactions. Greenberg, Harrison-Trainor, Patey, and Turetsky showed that we can always find some uniformity from an introenumerable set. We show that this can be reversed: we can construct an introenumerable set by patching up uniformity. This gives a rise to a new method of constructing a nontrivial introenumerable or introreducible set.
Paper Structure (14 sections, 55 theorems, 18 equations, 1 figure)

This paper contains 14 sections, 55 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation
  4. Main results of this paper
  5. Binary relations
  6. Majorenumerable and majorreducible sets
  7. Regressive and retraceable sets
  8. Co-immune regressive/retraceable sets
  9. Introenumerable and introreducible sets
  10. Co-c.e. introenumerable sets
  11. Question arising from Jockusch's work
  12. Building an introenumerable set
  13. Remaining separations
  14. Building a nontrivial regressive set

Key Result

Theorem 1.1

Uniformly major-T/e, major-T/e, uniformly intro-T/e, intro-T/e, regressive, retraceable, c.e., and computable are all distinct notions. Any two notions $P$, $Q$ from the following four are refinable to each other.

Figures (1)

  • Figure 1: The universe of introenumerable sets. The bottom four boxes (to the right of 'intro-T' label) are the introreducible sets, and the right six boxes (under the 'unif intro-e' label) are the uniformly introenumerable sets. Similarly, the four right-most boxes are the regressive sets, and the two right-most boxes are the c.e. sets. The intersection of the uniformly introenumerable boxes and the introreducible boxes is colored in gray, and it is exactly for the uniformly introreducible sets. The trapezoid with the thick lines is for the retraceable sets.

Theorems & Definitions (104)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2: Solovay
  • Corollary 2.3: Jockusch, Solovay
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • ...and 94 more