Index Sets of Universal Codes

TL;DR

This paper investigates the properties and complexity of universal code sets, establishing their connections to Medvedev reductions and proving high-level completeness results within the arithmetic hierarchy.

Contribution

It introduces a framework linking universal codes to Medvedev reductions and proves their complexity at various levels, including $ ext{Pi}_1^1$-completeness.

Findings

Proves completeness results at various arithmetic hierarchy levels.

Establishes a connection between universal codes and Medvedev reductions.

Shows the set of codes for certain Medvedev reductions is $ ext{Pi}_1^1$-complete.

Abstract

We examine sets of codes such that certain properties are invariant under the choice of oracle from a range of possible oracles and establish a connection between such codes and Medvedev reductions. In examing the complexity of such sets of \emph{universal codes}, we prove completeness results at various levels of the arithmetic hierarchy as well as two general theorems for obtaining $\Pi_1^1$-completeness for sets of universal codes. Among other corollaries, we show that the set of codes for Medvedev reductions of bi-immune sets to DNC functions is $\Pi_1^1$-complete.