The Minimum Failure Detector For Non-Local Tasks In Message-Passing Systems

Intuitively, a task is local if the output value of each process depends only on the process’ own input value, not on the input values of the other processes; a task is non-local otherwise. In this paper, we use the failure detector abstraction to determine the minimum information about failures that is necessary to solve non-local tasks in message-passing systems. More precisely, we show that there is a non-trivial failure detector, denoted FS∗, that is necessary to solve non-local tasks, i.e., FS∗ can be extracted from any failure detector that can be used to solve any non-local task in message-passing systems. We also show that FS∗ is the strongest failure detector with this property. So, intuitively, FS∗ is the greatest lower bound of the set of failure detectors that solve non-local tasks in message-passing systems. Even though FS∗ is quite weak, it is strong enough to solve a natural weakening of the well-known set agreement task, that we call weak set agreement. In fact, we show that FS∗ is the weakest failure detector to solve the weak set agreement task. Finally, we compare FS∗ to two closely related failure detectors, namely, L and anti-Ω, which are the weakest failure detectors to solve set agreement in message-passing and shared memory systems, respectively. We prove that anti-Ω is strictly weaker than FS∗ and FS∗ is strictly weaker than L, in message-passing systems.