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.

翻译：本文定义了一类新的分布式任务，称为SOS任务（输出集集合任务），由可产生的不同输出值集合$O$所定义。我们随后证明这类任务是可判定的：存在有效过程可判定任意SOS任务在$t$次异步崩溃条件下是否可解。决策规则如下：当$t=0$时所有SOS任务均可解。对于$t>0$，SOS任务可解当且仅当其SOS图$G=(O,\subset)$是连通的。该图中每个顶点为$O$中的输出集，两个顶点间存在边当且仅当一个输出集包含另一个。本研究结论的意外启示之一是：在没有有效性属性的条件下，对于$k>1$的$k$-集合一致问题可在任意$t \geq 0$次崩溃下求解，而仅在$k=1$（共识问题）时在$t>0$次崩溃下不可解。最后，我们研究了一类新型任务——$d$-分歧问题，要求系统始终产生$d$个不同的输出值，并证明其可实施性与调和级数相关。