An Overview of Synchronous Message-Passing and Topology

A slowly-growing number of computer scientists have found that ideas from topology can be used to analyze and understand problems in distributed computing. In this paper, we review one approach we have used in the past to write a succinct proof of the lower bound for the number of rounds needed to solve the k-set agreement problem in a synchronous, message-passing model of computation. The central idea in this approach is a simple combinatorial structure we call a pseudosphere, in which each process from a set of processes is independently assigned a value from a set of values. Pseudospheres have a number of nice combinatorial properties, but their principal interest lies in the observation that the global states that arise in the synchronous, message-passing model can be viewed as simple unions of pseudospheres, and the fact that topological properties of unions of pseudospheres are so easy to prove. We choose this work to review because it is a simple example of how we model distributed systems with topology, and because it is the basis of on-going work to simplify the proof of this result.