Minimal covers in the Weihrauch degrees

Abstract

In this paper, we study the existence of minimal covers and strong minimal covers in the Weihrauch degrees. We characterize when a problem $f$ is a minimal cover or strong minimal cover of a problem $h$. We show that strong minimal covers only exist in the cone below $\mathsf{id}$ and that the Weihrauch lattice above $\mathsf{id}$ is dense. From this, we conclude that the degree of $\mathsf{id}$ is first-order definable in the Weihrauch degrees and that the first-order theory of the Weihrauch degrees is computably isomorphic to third-order arithmetic.