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

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’’ ($\mathsf{D}\mathsf{S}$). We prove that $\mathsf{D}\mathsf{S}$ is strictly Weihrauch reducible to $\mathsf{B}\mathsf{S}$, 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 $\mathsf{B}\mathsf{S}$ and $\mathsf{D}\mathsf{S}$ have the same finitary and deterministic parts, confirming that $\mathsf{B}\mathsf{S}$ and $\mathsf{D}\mathsf{S}$ have very similar uniform computational strength. We prove that König’s lemma $\mathsf{K}\mathsf{L}$ and the problem ${\mathsf{w}\mathsf{L}\mathsf{i}\mathsf{s}\mathsf{t}}_{2^{\mathbb{N}},\le \omega }$ of enumerating a given non-empty countable closed subset of $2^{\mathbb{N}}$ are not Weihrauch reducible to $\mathsf{D}\mathsf{S}$ or $\mathsf{B}\mathsf{S}$, 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 $\mathsf{B}\mathsf{S}$ and $\mathsf{D}\mathsf{S}$ under the quotient with some known problems.