# Topological Characterization of Consensus in Distributed Systems

## TL;DR

This paper characterizes the solvability of consensus in distributed systems with faults using novel topological methods, extending classical topology to analyze execution spaces and explain algorithmic possibilities.

## Contribution

It introduces fault-aware topologies on execution spaces, providing a unified topological framework for understanding consensus solvability and existing impossibility results.

## Key findings

- Consensus solvability corresponds to disconnected sets in the new topologies.
- The approach explains existing algorithms and impossibility results topologically.
- Develops a new equivalence between strong and weak validity conditions.

## Abstract

We provide a complete characterization of both uniform and non-uniform deterministic consensus solvability in distributed systems with benign process and communication faults using point-set topology. More specifically, we non-trivially extend the approach introduced by Alpern and Schneider in 1985, by introducing novel fault-aware topologies on the space of infinite executions: the process-view topology, induced by a distance function that relies on the local view of a given process in an execution, and the minimum topology, which is induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions. Consensus is solvable in a given model if and only if the sets of admissible executions leading to different decision values is disconnected in these topologies. By applying our approach to a wide range of different applications, we provide a topological explanation of a number of existing algorithms and impossibility results and develop several new ones, including a general equivalence of the strong and weak validity conditions.