On the Decidability of Distributed Tasks with Output Sets under Asynchrony and Any Number of Crashes

TL;DR

This paper introduces SOS tasks, proves their decidability based on a graph connectivity criterion, and explores implications for k-set agreement and d-disagreement tasks in asynchronous distributed systems.

Contribution

It defines SOS tasks and provides an effective decision procedure for their solvability under crashes, revealing new insights into agreement problems.

Findings

SOS tasks are solvable iff their SOS graph is connected for t>0.

Without a validity property, k-set agreement is solvable under any crashes for k>1.

Implementability of d-disagreement relates to the harmonic series.

Abstract

In this paper, we define a new class of distributed tasks, called SOS tasks (for Set of Output Sets tasks), defined by the set $O$ of distinct output sets of values that can be produced. We then demonstrate that this class of tasks is decidable: there exists an effective procedure that determines whether any SOS task is solvable asynchronously under $t$ crashes. The decision rule is as follows. Every SOS task is solvable when $t=0$. For $t > 0$, an SOS task is solvable if and only if its SOS graph $G=(O,\subset)$ is connected. In this graph, each vertex is an output set in $O$, and two vertices are linked by an edge whenever one output set includes the other. One of the surprising implications of our results is that, without a validity property, $k$-set agreement is solvable under any number of crashes $t \geq 0$ for $k>1$, and unsolvable under $t >0$ crashes only for $k=1$ (consensus). Finally, we study a novel family of tasks called $d$-disagreement, which requires the system to always produce $d$ different output values, and we show that its implementability condition is related to the harmonic series.