Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generically Computable Linear Orderings

Wesley Calvert, Douglas Cenzer, David Gonzalez, Valentina Harizanov

TL;DR

The authors are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.

Abstract

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $Σ_β$ hierarchy. We focus on linear orderings. We show that at the $Σ_1$ level all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the $Σ_{α+2}$ level for any $α\inω_1^{ck}$ the set of linear orderings with generically or coarsely computable copies is $\mathbfΣ_1^1$-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.

Generically Computable Linear Orderings

TL;DR

The authors are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.

Abstract

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the hierarchy. We focus on linear orderings. We show that at the level all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the level for any the set of linear orderings with generically or coarsely computable copies is -complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.
Paper Structure (20 sections, 61 theorems, 57 equations)

This paper contains 20 sections, 61 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Background about Linear Orderings
  3. $\Sigma_\alpha$ Elementary Substructures of Linear Orderings
  4. The Block Relations
  5. Block Sizes and Particular Linear Ordering Constructions
  6. Linear Orderings with Densely Computable Copies
  7. Structural Criteria for Dense Computability
  8. The level $\Sigma_1$
  9. $\Sigma_2$ for Scattered Linear Orderings
  10. Hardness at the Level $\Sigma_2$
  11. Iterating up the Hierarchy
  12. Connections with Other Notions in Computable Structure Theory
  13. Additional Constructions and Questions Near Limit Levels
  14. Constructions of Individual Orderings
  15. Coarsely c.e. orderings that are not generically c.e.
  16. ...and 5 more sections

Key Result

theorem 3.5

Every linear ordering has a $\Sigma_1$-generically c.e. copy and a $\Sigma_1$-coarsely c.e copy.

Theorems & Definitions (124)

  • definition 1.1
  • definition 1.2
  • definition 1.3
  • definition 1.4
  • definition 1.5
  • theorem 3.5
  • theorem 3.7
  • theorem 3.13
  • theorem 3.21
  • proposition 4.2
  • ...and 114 more