A jump operator on the Weihrauch degrees

TL;DR

This paper introduces a new jump operator called the totalizing jump in the Weihrauch degrees, addressing a longstanding question by defining an explicit operator that induces an injective endomorphism in the lattice.

Contribution

It provides the first explicit construction of a jump operator in the Weihrauch lattice, using the total continuation, and analyzes its algebraic properties.

Findings

The totalizing jump induces an injective endomorphism of the Weihrauch degrees.

It can be characterized via the total continuation operator.

The operator's behavior on key problems is thoroughly analyzed.

Abstract

A partial order $(P,\le)$ admits a jump operator if there is a map $j\colon P \to P$ that is strictly increasing and weakly monotone. Despite its name, the jump in the Weihrauch lattice fails to satisfy both of these properties: it is not degree-theoretic and there are functions $f$ such that $f\equiv_{\mathrm{W}} f'$. This raises the question: is there a jump operator in the Weihrauch lattice? We answer this question positively and provide an explicit definition for an operator on partial multi-valued functions that, when lifted to the Weihrauch degrees, induces a jump operator. This new operator, called the totalizing jump, can be characterized in terms of the total continuation, a well-known operator on computational problems. The totalizing jump induces an injective endomorphism of the Weihrauch degrees. We study some algebraic properties of the totalizing jump and characterize its behavior on some pivotal problems in the Weihrauch lattice.