Easy impossibility proofs for k-set agreement in message passing systems

Despite of being quite similar (agreement) problems, 1-set agreement (consensus) and general k-set agreement require surprisingly different techniques for proving the impossibility in asynchronous systems with crash failures: Rather than the relatively simple bivalence arguments as in the impossibility proof for consensus in the presence of a single crash failure, known proofs for the impossibility of k-set agreement in shared memory systems with f ≥ k > 1 crash failures use algebraic topology or a variant of Sperner's Lemma. In this paper, we present a generic theorem for proving the impossibility of k-set agreement in various message passing settings, which is based on a reduction to the consensus impossibility in a certain subsystem resulting from a partitioning argument. We demonstrate the broad applicability of our result by exploring the possibility/impossibility border of k-set agreement in several message-passing system models: (i) asynchronous systems with crash failures, (ii) partially synchronous processes with (initial) crash failures, and, most importantly, (iii) asynchronous systems augmented with failure detectors. In (i), (ii), and (iii), the impossibility part is an instantiation of our main theorem, whereas the possibility of achieving k-set agreement in (ii) follows by generalizing the consensus algorithm for initial crashes by Fisher, Lynch and Patterson. In (iii), applying our technique reveals the exact border for the parameter k where k-set agreement is solvable with the failure detector class (∑_k, Ω_k) _1≤k≤n-1 of Bonnet and Raynal. As ∑_k was shown to be necessary for solving k-set agreement, this result yields new insights on the quest for the weakest failure detector.