Relationships between computability-theoretic properties of problems

Abstract

A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set $A$ and a computable instance of a problem $\mathsf{P}$, to find a solution relative to which $A$ is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of $\omega$ cones, of 1 hyperimmunity or of 1 non-$\Sigma^0_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non-$\Sigma^0_1$ definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting.