On the weakest failure detector ever

Many problems in distributed computing are impossible when no information about process failures is available. It is common to ask what information about failures is necessary and sufficient to circumvent some specific impossibility, e.g., consensus, atomic commit, mutual exclusion, etc. This paper asks what information about failures is needed to circumvent any impossibility and sufficient to circumvent some impossibility. In other words, what is the minimal yet non-trivial failure informatio. We present an abstraction, denoted Υ, that provides very little failure information. In every run of the distributed system, Υ eventually informs the processes that some set of processes in the system cannot be the set of correct processes in that run. Although seemingly weak, for it might provide random information for an arbitrarily long period of time, and it only excludes one possibility of correct set among many, Υ still captures non-trivial failure information. We show that Υ is sufficient to circumvent the fundamental wait-free set-agreement impossibility. While doing so, we (a) disprove previous conjectures about the weakest failure detector to solve set-agreement and we (b) prove that solving set-agreement with registers is strictly weaker than solving n+1-process consensus using n-process consensus. We prove that Υ is, in a precise sense, minimal to circumvent any wait-free impossibility. Roughly, we show that Υ is the weakest eventually stable failure detect or to circumvent any wait-free impossibility. Our results are generalized through an abstraction Υf that we introduce and prove necessary to solve any problem that cannot be solved in an f-resilient manner, and yet sufficient to solve f-resilient f-set-agreement.