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

2024-01-22

Conference on Computability in Europe

AI Summary

This paper investigates the Weihrauch computational strength of descending sequences (DS) in non-well-founded linear orders and bad sequences (BS) in non-well-quasi-orders. Using Weihrauch reducibility analysis, the parallel quotient operator construction, and first-order partial separation techniques, it establishes—rigorously for the first time—that DS is strictly weaker than BS (i.e., DS <ₐ BS), correcting prior erroneous claims. It further shows that DS and BS are equivalent with respect to finite and deterministic parts. Crucially, the paper refutes Weihrauch reducibility of König’s Lemma (KL) and the problem of enumerating countable closed subsets of ℕ^ℕ (wList₂ᴺ) to either DS or BS, thereby resolving two long-standing open problems. Additionally, it develops a comprehensive theoretical framework characterizing the behavior of BS and DS under quotient operations, precisely delineating their relative computational positions within the Weihrauch lattice.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Agreement, Argumentation & Negotiation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 $mathsf{wList}_{2^{mathbb{N}},leqomega}$ 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.

Problem

• Compare Weihrauch degrees of finding bad sequences in non-well quasi orders and descending sequences in ill-founded linear orders
• Separate first-order parts of BS and DS to prove strict Weihrauch reducibility
• Investigate non-reducibility of König's lemma and enumerating countable closed subsets to BS or DS

Innovation

• Weihrauch reducibility analysis of BS and DS
• Separation of first-order parts for BS and DS
• Proof of non-reducibility involving König's lemma