From Geometric Semantics to Asynchronous Computability

TL;DR: By constructing a one-to-one relationship between simplices of the protocol complex and dihomotopy classes of dipaths in the latter semantics, a connection between these two geometric approaches to distributed computing: protocol complexes and directed algebraic topology is described.

Abstract: 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 dihomotopy classes of dipaths 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.