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We determine the weakest failure detectors to solve several fundamental problems in distributed message-passing systems, for all environments -i.e., regardless of the number and timing of crashes. The problems that we consider are: implementing an atomic register, solving consensus, solving quittable consensus (a variant of consensus in which processes have the option to decide 'quit' if a failure occurs), and solving non-blocking atomic commit.

Abstract. In the set-agreement problem, n processes seek to agree on at most n−1 different values. This paper determines the weakest failure detector to solve this problem in a message-passing system where processes may fail by crashing. This failure detector, called the Loneliness detector and denoted L, outputs one of two values, "true" or "false" such that:(1) there is at least one process where L outputs always "false", and (2) if only one process is correct, L eventually outputs "true" at this process.