Topological Characterization of Consensus in Distributed Systems

TL;DR

This work non-trivially extends 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, induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions.

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.

Figures (6)

• Figure 1: Comparison of the combinatorial topology approach and the point-set topology approach: The combinatorial topology approach (left) studies sequences of increasingly refined spaces in which the objects of interest are simplices (corresponding to configurations). The point-set topology approach (right) studies a single space in which the objects of interest are executions (i.e., infinite sequences of configurations).
• Figure 2: Comparison of the $p$-view and common-prefix metric. The first three configurations of each of the two executions $\gamma$ and $\delta$ with three processes and two different possible local states (dark blue and light yellow) are depicted. We have $d_{\max}(\gamma,\delta) = d_3(\gamma,\delta) = 1$ and $d_2(\gamma,\delta) = 1/2$.
• Figure 3: Examples of two connected components of the decision sets $\Sigma_0=\Sigma_{\gamma_0}\cup \Sigma_{\gamma_0'}$ and $\Sigma_1=\Sigma_{\gamma_1}\cup \Sigma_{\gamma_1'}$ for consensus under a limit-closed message adversary. contain all their limit points (marked by $\times$) and have a distance $>0$ by cor:closeddecsetscompact.
• Figure 4: Examples of two connected components of the decision sets $\Sigma_0=\Sigma_{\gamma_0}\cup \Sigma_{\gamma_0'}$ and $\Sigma_1=\Sigma_{\gamma_1}\cup \Sigma_{\gamma_1'}$ for a non-compact message adversary. They are not closed in $\mathcal{C}^{\omega}$ and may have distance 0; common limit points (like for $\Sigma_{\gamma_0}$ and $\Sigma_{\gamma_1}$, marked by $\times$) must hence be excluded by Corollary \ref{['cor:consensusseparation']}.
• Figure 5: Example of a process-time graph prefix $PTG^3$ of a lock-step execution at time $t=3$, for $n=3$ processes and initial values $x=(1,0,1)$. Process $1$'s view $V_{1}(PT^2)$ is highlighted in bold green.

Theorems & Definitions (51)

• definition 1: Non-uniform and uniform consensus
• lemma 1
• lemma 2
• lemma 3: Pseudometric $d_p$
• lemma 4
• lemma 5
• definition 2: $v$-valent execution
• theorem 1: Characterization of uniform consensus
• theorem 2: Characterization of non-uniform consensus
• definition 3: Distance of sets