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. A discussion of a restricted type of reduction has shown that there are no extension-based proofs of the impossibility of wait-free protocols for some other distributed computing problems. We extend the previous result to general reductions that allow multiple instances of tasks. The techniques used in the previous work are designed for certain tasks, such as the $(n,k)$-set agreement task. We give a necessary and sufficient condition for general colorless tasks to have no extension-based proofs of the impossibility of wait-free protocols, and show that different types of extension-based proof are equivalent in power for colorless tasks. Using this necessary and sufficient condition, the result about reductions can be understood from a topological perspective.

翻译：基于扩展的证明（extension-based proofs）概念建模了分布式计算中广泛使用的值论证（valency argument）思想。这类证明的能力存在局限：已有研究表明，对于在$n > k \geq 2$个进程间实现$(n,k)$-集合共识的无等待协议的不可行性，不存在基于扩展的证明。针对一种受限归约类型的讨论显示，对于其他若干分布式计算问题，也不存在关于无等待协议不可行性的基于扩展的证明。本文将先前结果推广至允许多任务实例的通用归约。此前工作中使用的技术针对特定任务（如$(n,k)$-集合共识任务）而设计。我们给出了通用无色任务不存在无等待协议不可行性之基于扩展证明的充要条件，并证明对于无色任务，不同类型的基于扩展证明在能力上等价。利用这一充要条件，可以从拓扑视角理解关于归约的结论。