Easy Impossibility Proofs for k-Set Agreement in Message Passing Systems

TL;DR

This paper introduces a simple, generic theorem for proving the impossibility of k-set agreement in message passing systems, simplifying previous complex proofs and exploring the boundaries of solvability across various models.

Contribution

The paper presents a unified, reduction-based theorem for impossibility proofs of k-set agreement, broadening understanding across multiple message-passing system models.

Findings

Theorem applies to asynchronous crash-failure systems, partially synchronous models, and failure detector-augmented systems.

Exact boundary for k-set agreement solvability with specific failure detectors (Sigma_k, Omega_k).

New insights into the minimal failure detector requirements for k-set agreement.

Abstract

Despite of being quite similar agreement problems, consensus and general k-set agreement require surprisingly different techniques for proving the impossibility in asynchronous systems with crash failures: Rather than relatively simple bivalence arguments as in the impossibility proof for consensus (= 1-set agreement) in the presence of a single crash failure, known proofs for the impossibility of k-set agreement in systems with at least 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 simple reduction to the consensus impossibility in a certain subsystem. 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 (iii) asynchronous systems augmented with failure detectors. In (i) and (ii), the impossibility part is just 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 yields the exact border for the parameter k where k-set agreement is solvable with the failure detector class (Sigma_k,Omega_k), for (1<= k<= n-1), of Bonnet and Raynal. Considering that Sigma_k was shown to be necessary for solving k-set agreement, this result yields new insights on the quest for the weakest failure detector.