The computational content of multidimensional discontinuity

TL;DR

This paper extends the analysis of discontinuity in computational problems from one dimension to multiple dimensions within the Weihrauch degrees framework, revealing strict hierarchies based on the complexity of discontinuous steps.

Contribution

It generalizes previous single-dimensional results to multiple dimensions, establishing new hierarchies in the Weihrauch degrees related to discontinuity complexity.

Findings

Established strict hierarchies in Weihrauch degrees

Ordered problems by the richness of truth-tables affecting discontinuity

Extended previous work from one to multiple dimensions

Abstract

The Weihrauch degrees are a tool to gauge the computational difficulty of mathematical problems. Often, what makes these problems hard is their discontinuity. We look at discontinuity in its purest form, that is, at otherwise constant functions that make a single discontinuous step along each dimension of their underlying space. This is an extension of previous work of Kihara, Pauly, Westrick from a single dimension to multiple dimensions. Among other results, we obtain strict hierarchies in the Weihrauch degrees, one of which orders mathematical problems by the richness of the truth-tables determining how discontinuous steps influence the output.