Relating $\mathcal{L}$-Resilience and Wait-Freedom via Hitting Sets

Abstract

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.

We show that the ability of arbitrary collection of live sets $\mathcal{L}$ to solve distributed tasks is tightly related to the minimum hitting set of $\mathcal{L}$, a minimum cardinality subset of processes that has a non-empty intersection with every live set. Thus, finding the computing power of $\mathcal{L}$ is NP-complete.

For the special case of colorless tasks that allow participating processes to adopt input or output values of each other, we use a simple simulation to show that a task can be solved $\mathcal{L}$ -resiliently if and only if it can be solved (h − 1)-resiliently, where h is the size of the minimum hitting set of $\mathcal{L}$.

For general tasks, we characterize $\mathcal{L}$-resilient solvability of tasks with respect to a limited notion of weak solvability: in every execution where all processes in some set in $\mathcal{L}$ are correct, outputs must be produced for every process in some (possibly different) participating set in $\mathcal{L}$. Given a task T, we construct another task $T_{\mathcal{L}}$ such that T is solvable weakly $\mathcal{L}$-resiliently if and only if $T_{\mathcal{L}}$ is solvable weakly wait-free.