On Set Consensus Numbers

Abstract

We propose a complete characterization of a large class of distributed tasks, with respect to a weakened solvability notion called weak termination. A task is weak-termination solvable if there is an algorithm by which at least one process outputs.

The proposed categorization of tasks is based on the weakest failure detectors needed to solve them. We show that every task $\mathcal{T}$ in the considered class is equivalent (in the failure detector sense) to some form of set agreement, and thus its solvability with weak termination is completely characterized by its set consensus number: the maximal integer k such that $\mathcal{T}$ can be (weak-termination) solved using read-write registers and k-set agreement objects.

The characterization goes through showing that $\neg{\it \Omega}_k$, recently shown to be the weakest failure detector for the task of k-set agreement, is necessary to solve any task that is k-resilient impossible.