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.

翻译：本文研究问题$\mathsf{DS}$的Weihrauch度，该问题要求通过给定的病态线性序寻找一条无限降链，这一性质与通过给定非良拟序寻找坏序列的问题$\mathsf{BS}$共享。我们表明，尽管$\mathsf{DS}$难以求解（其可计算输入不存在超算术解），但其在均匀计算强度方面相当弱。为精确阐述后者，我们引入了Weihrauch度确定性部分的概念。随后，我们通过考虑$\boldsymbol{\Gamma}$-表征集序（其中$\boldsymbol{\Gamma}$为Borel点类或$\boldsymbol{\Delta}^1_1$、$\boldsymbol{\Sigma}^1_1$、$\boldsymbol{\Pi}^1_1$），推广了$\mathsf{DS}$与$\mathsf{BS}$。我们研究了所得到的$\mathsf{DS}$-层级与$\mathsf{BS}$-层级问题，并与（有效）Baire层级进行比较，证明它们在任意有限层级均不坍缩。