Algebraic properties of the first-order part of a problem

TL;DR

This paper investigates the algebraic properties of the first-order part of computational problems within the Weihrauch lattice, introducing new operators and characterizing the first-order parts of various problems.

Contribution

It introduces the unbounded finite parallelization operator and explores its role in understanding the first-order part of parallelizable problems.

Findings

The first-order part can be characterized using the new unbounded finite parallelization operator.

Results facilitate explicit characterization of the first-order parts of several known problems.

The study enhances understanding of the algebraic structure of the Weihrauch lattice.

Abstract

In this paper we study the notion of first-order part of a computational problem, first introduced by Dzhafarov, Solomon, and Yokoyama, which captures the "strongest computational problem with codomain $\mathbb{N}$ that is Weihrauch reducible to $f$". This operator is very useful to prove separation results, especially at the higher levels of the Weihrauch lattice. We explore the first-order part in relation with several other operators already known in the literature. We also introduce a new operator, called unbounded finite parallelization, which plays an important role in characterizing the first-order part of parallelizable problems. We show how the obtained results can be used to explicitly characterize the first-order part of several known problems.