Weakest failure detectors via an egg-laying simulation ( Preliminary Version )

In the k-set agreement task, n processes propose values, and have to decide on at most k of these values. In particular, consensus is 1-set agreement. In PODC 2008 Zieliński showed that the anti−Ω failure detector is necessary and sufficient to solve (n − 1)-set agreement in an asynchronous read/write shared memory system where at most t processes can fail by crashing, t = n− 1. In this paper it is shown that anti−Ω is the weakest failure detector to solve t-set agreement in a t-resilient asynchronous distributed system. Each query to anti−Ω returns a set S of process ids, |S| = n − t, such that some correct process eventually never appears in such a set S; thus, anti−Ωn−1 = anti−Ω , and anti−Ω = Ω. Actually, the paper shows a stronger result: Any failure detector that can be used to solve T is at least as strong as anti−Ω, for any agreement task T that has no t-resilient solution. The previous results are obtained through a variant of Zieliński’s technique, that simplifies some of the arguments and introduces an egg-laying simulation of independent interest. The simulation provides the first detailed analysis of immediate snapshot executions in a failure detector enriched environment. It provides a simple proof that Ω is the weakest failure detector to solve consensus when t = 1.