L-Resilient Adversaries and Hitting Sets

The condition of t-resilience stipulates that an n-process program is only obliged to make progress when at least n− t processes are correct. Put another way, the live sets, the collection of process sets such that progress is required if all the processes in one of these sets are correct, are all sets with at least n − t processes. In this paper we study what happens when the live sets are any arbitrary collection of sets L. We show that the power of L to solve distributed tasks is tightly related to the minimum hitting set of L, a minimum cardinality subset of processes that has a non-empty intersection with every live set. Thus, a necessary condition to make progress in the presence of L is that at least one member of the set is correct. For the special case of colorless tasks that allow participating processes to adopt input or output values of each other, we show that the set of tasks that can be solved L-resiliently is exactly captured by the size of the minimum hitting set of L. For general tasks, we characterize L-resilient solvability of tasks with respect to a limited notion of weak solvability: in every execution where all processes in some set in L are correct, outputs must be produced for every process in some (possibly different) participating set in L. Given a task T , we construct another task T ′ such that T is solvable weakly L-resiliently if and only if T ′ is solvable weakly wait-free.