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Finding descending sequences through ill-founded linear orders

Jun Le Goh, Arno Pauly, Manlio Valenti

TL;DR

This work studies the uniform computational content of finding descending sequences in ill-founded linear orders (${\rm DS}$) and its bad-sequence counterpart (${\rm BS}$) through Weihrauch reducibility. It introduces the deterministic part ${\rm Det}$, analyzes its interaction with codomain spaces, and proves that ${\rm Det}({\rm DS}) \equiv_W {\rm lim}$, while DS remains hard in a nonuniform sense. By generalizing to ${\boldsymbol\Gamma}$-presented orders, the authors establish non-collapsing DS-/BS-hierarchies at finite levels and relate these to classical hierarchies such as ${\rm lim}^{(n)}$, ${\rm C}_{\mathbb{N}^{\mathbb{N}}}$, and ${\boldsymbol{\Pi^1_1{\text{-CA}}} }$. The results clarify the subtle balance between uniform computational strength and nonuniform computability in well-/ill-founded contexts and situate DS/BS within the broader Weihrauch lattice, with several open questions highlighted for future work.

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 $\boldsymbolΓ$-presented orders, where $\boldsymbolΓ$ is a Borel pointclass or $\boldsymbolΔ^1_1$, $\boldsymbolΣ^1_1$, $\boldsymbolΠ^1_1$. 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.

Finding descending sequences through ill-founded linear orders

TL;DR

This work studies the uniform computational content of finding descending sequences in ill-founded linear orders () and its bad-sequence counterpart () through Weihrauch reducibility. It introduces the deterministic part , analyzes its interaction with codomain spaces, and proves that , while DS remains hard in a nonuniform sense. By generalizing to -presented orders, the authors establish non-collapsing DS-/BS-hierarchies at finite levels and relate these to classical hierarchies such as , , and . The results clarify the subtle balance between uniform computational strength and nonuniform computability in well-/ill-founded contexts and situate DS/BS within the broader Weihrauch lattice, with several open questions highlighted for future work.

Abstract

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

Paper Structure

This paper contains 17 sections, 63 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of our results
  3. Structure of the paper
  4. Background
  5. The deterministic part of a problem
  6. Impact of the codomain space
  7. The deterministic part and the first-order part
  8. Interaction with other operations on Weihrauch degrees
  9. Previous appearances in the literature
  10. Finding descending sequences
  11. The uniform strength of DS
  12. Combinatorial principles on linear orders
  13. Relations with Ramsey theorems
  14. Presentation of orders
  15. Gamma0k-DS and Gamma0k-BS
  16. ...and 2 more sections

Key Result

Theorem 2.1

No Environment Found If $f:\subseteq {\mathbb{N}^\mathbb{N}} \rightrightarrows \mathbf{X}$ is Weihrauch reducible to $\mathsf{UC}_{\mathbb{N}^\mathbb{N}}$, then for every $x\in\operatorname{dom}(f)$, $f(x)$ contains some $y$ hyperarithmetical relative to $x$.

Figures (2)

  • Figure 1: An overview of some parts of the Weihrauch lattice. The solid frame collects the degrees belonging to the lower cone of $\mathsf{DS}$, while the dashed frame collects principles that are not Weihrauch reducible to $\mathsf{DS}$. The only principle shown which is above $\mathsf{DS}$ is $\mathsf{C}_{\mathbb{N}^\mathbb{N}}$. We do not know whether $\mathsf{KL}$ is reducible to $\mathsf{DS}$.
  • Figure 2: Diagram presenting the relations between the various generalizations of $\mathsf{DS}$.

Theorems & Definitions (130)

  • Theorem 2.1: KMP20
  • Definition 2.2
  • Proposition 2.3: DSYFirstOrder
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • ...and 120 more