Colorless Tasks and Extension-Based Proofs

TL;DR

The paper develops a complete characterization of when colorless tasks admit extension-based impossibility proofs in the NIIS model. It introduces an adversarial strategy anchored by canonical neighbors and chromatic subdivisions to construct compatible partial protocols, proving a necessary-and-sufficient condition for nonexistence of extension-based proofs. The results generalize prior findings for (n,k)-set agreement to a broad class of colorless tasks and provide a unified framework for analyzing extensions, finalization, and assignment queries. This advances theoretical understanding of asynchronous computability and offers practical insights for topological representations of distributed protocols. The work also highlights potential extensions to colored tasks and lays groundwork for future exploration of canonical-neighbor techniques in extension-based proof contexts.

Abstract

The concept of extension-based proofs models the idea of a valency argument, which is widely used in distributed computing. Extension-based proofs are limited in power: it has been shown that there is no extension-based proof of the impossibility of a wait-free protocol for $(n,k)$-set agreement among $n > k \geq 2$ processes. There are only a few tasks that have been proven to have no extension-based proof of the impossibility, since the techniques in these works are closely related to the specific task. We give a necessary and sufficient condition for colorless tasks to have no extension-based proofs of the impossibility of wait-free protocols in the NIIS model. We introduce a general adversarial strategy decoupled from any concrete task specification. In this strategy, some properties of the chromatic subdivision that is widely used in distributed computing are proved.

Figures (10)

• Figure 1: The standard chromatic subdivision of a 2-dimensional complex
• Figure 2: A non-uniform chromatic subdivision of a 2-dimensional complex
• Figure 3: The protocol complexes $\mathbb{F}_{0}(U)$ and $\mathbb{F}_{2}(U)$ with respect to $U = \{(p_{0}, 0)\}$
• Figure 4: An example of our implementation of canonical neighbors when $r_{m} = 2$
• Figure 5: An example of $s_{0}, s_{1} \cdots s_{e}$ when $r_{m} = 3$

Theorems & Definitions (30)

• Theorem 1: Asynchronous computability theorem
• Theorem 2: Theorem \ref{['the:partial_info_theorem']}
• Theorem 3: Theorem \ref{['the:adversary_finalize_after_the_first_round_theorem']}
• Theorem 4: Theorem \ref{['the:assignment_queries_no_more_power']}
• Theorem 5: Theorem \ref{['the:adversary_finalize_theorem']}
• Lemma 6
• Lemma 7
• Theorem 8
• Lemma 9
• Lemma 10