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Minimal covers in the Weihrauch degrees

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

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.

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 is a minimal cover or strong minimal cover of a problem . We show that strong minimal covers only exist in the cone below and that the Weihrauch lattice above is dense. From this, we conclude that the degree of 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.
Paper Structure (4 sections, 16 theorems, 6 equations)

This paper contains 4 sections, 16 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our main theorems
  4. Proof of the main theorems

Key Result

Theorem 1.1

For every $A<_\mathrm{M} B$, $B$ is a minimal cover of $A$ iff where $P \wedge Q:=({0}){\raisebox{.9ex}{$$\smallfrown} } P \cup ({1}){\raisebox{.9ex}{$$\smallfrown} } Q$ is the meet in the Medvedev degrees.

Theorems & Definitions (26)

  • Theorem 1.1: Dyment Dyment1976
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Corollary 1.7
  • Theorem 1.8
  • proof
  • ...and 16 more