On the first-order parts of problems in the Weihrauch degrees

TL;DR

This paper introduces the concept of the first-order part of problems within the Weihrauch degrees, establishing its properties and characterizing it for well-known problems, thereby advancing the understanding of problem reducibility.

Contribution

It defines the first-order part of a problem in the Weihrauch degrees, proves its existence, and characterizes it for several classical problems.

Findings

The first-order part is always well-defined.

Basic properties of the first-order part are established.

Characterizations are provided for several known problems.

Abstract

We introduce the notion of the \emph{first-order part} of a problem in the Weihrauch degrees. Informally, the first-order part of a problem $\mathsf{P}$ is the strongest problem with codomaixn $\omega$ that is Weihrauch reducible to $\mathsf{P}$. We show that the first-order part is always well-defined, examine some of the basic properties of this notion, and characterize the first-order parts of several well-known problems from the literature.