Finding descending sequences through ill-founded linear orders

TL;DR

This paper analyzes the computational complexity of finding infinite descending sequences in ill-founded linear orders, revealing that these problems are computationally weak despite their difficulty, and explores their hierarchy within descriptive set theory.

Contribution

It introduces the notion of the deterministic part of a Weihrauch degree and generalizes the problems to various Borel and projective classes, establishing their non-collapse in hierarchies.

Findings

$ extsf{DS}$ is computationally weak despite being hard to solve.

The $ extsf{DS}$ and $ extsf{BS}$ hierarchies do not collapse at any finite level.

The hierarchy comparison with the Baire hierarchy shows a rich structure of these problems.

Abstract

In this work we investigate the Weihrauch degree of the problem $\mathsf{DS}$ of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem $\mathsf{BS}$ of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf{DS}$, despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf{DS}$ and $\mathsf{BS}$ by considering $\boldsymbol{\Gamma}$-presented orders, where $\boldsymbol{\Gamma}$ is a Borel pointclass or $\boldsymbol{\Delta}^1_1$, $\boldsymbol{\Sigma}^1_1$, $\boldsymbol{\Pi}^1_1$. We study the obtained $\mathsf{DS}$-hierarchy and $\mathsf{BS}$-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.