We show that the protocol complex formalization of fault-tolerant protocols can be directly derived from a suitable semantics of the underlying synchronization and communication primitives, based on a geometrization of the state space. By constructing a one-to-one relationship between simplices of the protocol complex and (di)homotopy classes of (di)paths in the latter semantics, we describe a connection between these two geometric approaches to distributed computing: protocol complexes and directed algebraic topology. This is exemplified on atomic snapshot, iterated snapshot and layered immediate snapshot protocols, where a well-known combinatorial structure, interval orders, plays a key role. We believe that this correspondence between models will extend to proving impossibility results for much more intricate fault-tolerant distributed architectures.

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关于异步可计算性的几何和组合视图

我们表明，基于状态空间的几何化，容错协议的协议复杂形式化可以直接从底层同步和通信原语的合适语义导出。通过在后一种语义中构建协议复合体的单纯形和（双）路径的（双）同伦类之间的一对一关系，我们描述了分布式计算的这两种几何方法之间的联系：协议复合体和有向代数拓扑. 这以原子快照、迭代快照和分层即时快照协议为例，其中众所周知的组合结构、间隔顺序起着关键作用。