Index Sets of Universal Codes

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 universal codes, we prove completeness results at various levels of the arithmetic hierarchy as well as two general theorems for obtaining Π 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 Π 1 1 -complete.