The condition of <i>t</i>-resilience stipulates that an <i>n</i>-process program is only obliged to make progress when at least <i>n-t</i> processes are correct. Put another way, the <i>live sets</i>, the collection of process sets such that progress is guaranteed if at least one of the sets is correct, are all sets with at least <i>n-t</i> processes.
Given an arbitrary collection of live sets <i>L</i>, what distributed tasks are solvable? We show that the power of <i>L</i> to solve tasks is tightly related to the <i>L</i>, <i>minimum hitting set</i>, of <i>L</i>, a minimum cardinality subset of processes that has a non-empty intersection with every live set. A necessary condition to make progress in the presence of <i>L</i> is that at least one member of the set is correct. Thus, finding the computing power of <i>L</i>, is <i>NP</i>-complete.
For the special case of <i>colorless,</i> tasks that allow every process to adopt an input or output value of any other process, we show that the set of tasks that an <i>L</i>-resilient adversary can solve is exactly captured by the size of its minimum hitting set.
For general tasks, we characterize <i>L</i>-resilient solvability of tasks with respect to a limited notion of <i>weak</i> solvability (which is however stronger than colorless solvability). Given a task <i>T</i>, we construct another task <i>T'</i> such that <i>T</i> is solvable weakly <i>L</i>-resiliently if and only if <i>T'</i> is solvable weakly wait-free.
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