Index Sets of Universal Codes
Achilles A. Beros, Konstantinos A. Beros

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 -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 -complete.
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Taxonomy
TopicsCoding theory and cryptography · semigroups and automata theory · Computability, Logic, AI Algorithms
