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arXiv · 2311.12676

Minimal covers in the Weihrauch degrees

Steffen Lempp·Joseph S. Miller·Arno Pauly·Mariya I. Soskova·Manlio Valenti
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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.

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Steffen Lempp, Joseph S. Miller, Arno Pauly, Mariya I. Soskova, Manlio Valenti. 2023-11-21. Minimal covers in the Weihrauch degrees. https://doi.org/10.1090/proc%2F16952

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