The weakness of finding descending sequences in ill-founded linear orders

Abstract

We explore the Weihrauch degree of the problems ``find a bad sequence in a non-well quasi order'' ($\mathsf{BS}$) and ``find a descending sequence in an ill-founded linear order'' ($\mathsf{DS}$). We prove that $\mathsf{DS}$ is strictly Weihrauch reducible to $\mathsf{BS}$, correcting our mistaken claim in [arXiv:2010.03840]. This is done by separating their respective first-order parts. On the other hand, we show that $\mathsf{BS}$ and $\mathsf{DS}$ have the same finitary and deterministic parts, confirming that $\mathsf{BS}$ and $\mathsf{DS}$ have very similar uniform computational strength. We prove that K\"onig's lemma $\mathsf{KL}$ and the problem $\mathsf{wList}_{2^{\mathbb{N}},\leq\omega}$ of enumerating a given non-empty countable closed subset of $2^{\mathbb{N}}$ are not Weihrauch reducible to $\mathsf{DS}$ or $\mathsf{BS}$, resolving two main open questions raised in [arXiv:2010.03840]. We also answer the question, raised in [arXiv:1804.10968], on the existence of a ``parallel quotient'' operator, and study the behavior of $\mathsf{BS}$ and $\mathsf{DS}$ under the quotient with some known problems.