Finding descending sequences through ill-founded linear orders

10/08/2020

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by   Jun Le Goh, et al.

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In this work we investigate the Weihrauch degree of the problem 𝖣𝖲 of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem 𝖡𝖲 of finding a bad sequence through a given non-well quasi-order. We show that 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 𝖣𝖲 and 𝖡𝖲 by considering Γ-presented orders, where Γ is a Borel pointclass or Δ^1_1, Σ^1_1, Π^1_1. We study the obtained 𝖣𝖲-hierarchy and 𝖡𝖲-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.