The gap in circumventing the impossibility of consensus

Abstract

The impossibility of reaching deterministic consensus in an asynchronous and crash prone system was established for a weak variant of the problem, usually called weak consensus, where a set of processes need to decide on a common value in {0, 1}, so that both 0 and 1 are possible decision values. On the other hand, approaches to circumventing the impossibility focused on a stronger variant of the problem, called consensus, where the processes need to decide on one of the values they initially propose (0 or 1). This paper studies the computational gap between the two problems. We show that any set of deterministic object types that, combined with registers, implements weak consensus, also implements consensus. Then we exhibit a non-deterministic type that implements weak consensus, among any number of processes, but, combined with registers, cannot implement consensus even among two processes. In modern terminology, this type has consensus power 1 and weak consensus power ∞.

The failure detector abstraction

“…This contrasts a weaker variant of consensus where we would require only that algorithm A has a run where 1 is decided and a run where 0 is decided. Considering such a weak variant impacts the result as pointed out by and Guerraoui and Kuznetsov [2008]. -Finally, it is important to recall that CHT is only the necessary part of the proof that Ω is the weakest failure detector for consensus.…”

“…The top level (level n−1) is populated by universal tasks (e.g., consensus): if a failure detector solves a universal task, then it solves any task. The weakest failure detector to solve a universal task is Ω Guerraoui and Kuznetsov 2008]. Now level ( = 1, .…”