Problem
Research questions and friction points this paper is trying to address.
distributed tasks
decidability
asynchrony
crash failures
output sets
On the Decidability of Distributed Tasks with Output Sets under Asynchrony and Any Number of Crashes
🤖 AI Summary
This study addresses the solvability of a class of distributed tasks—referred to as SOS tasks—defined by sets of output values in asynchronous systems subject to an arbitrary number of crash failures. By constructing an inclusion graph over output sets (the SOS graph) and leveraging tools from combinatorial topology and graph theory, the work establishes, for the first time, that the SOS task class is decidable under the asynchronous crash-failure model. The main contributions include a necessary and sufficient condition for solvability: SOS tasks are always solvable when the number of failures \( t = 0 \); for \( t > 0 \), they are solvable if and only if the SOS graph is connected. Additionally, the paper reveals the counterintuitive result that \( k \)-set agreement is always solvable for \( k > 1 \) in the absence of validity constraints, and it elucidates a connection between the realizability of \( d \)-disagreement tasks and 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
SOS tasks
decidability
asynchronous distributed computing
k-set agreement
d-disagreement
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