On set consensus numbers

Abstract

It is conjectured that the only way a failure detector (FD) can help solving n-process tasks is by providing k-set consensus for some k ∈ {1, . . . , n} among all the processes. It was recently shown by Zieliński that any FD that allows for solving a given n-process task that is unsolvable read-write wait-free, also solves (n − 1)-set consensus. In this paper, we provide a generalization of Zieliński's result. We show that any FD that solves a colorless task that cannot be solved read-write k-resiliently, also solves k-set consensus. More generally, we show that every colorless task T can be characterized by its set consensus number: the largest k ∈ {1, . . . , n} such that T is solvable (k − 1)-resiliently. A task T with set consensus number k is, in the failure detector sense, equivalent to k-set consensus, i.e., a FD solves T if and only if it solves k-set consensus. As a corollary, we determine the weakest FD for solving k-set consensus in every environment, i.e., for all assumptions on when and where failures might occur.

Asynchronous Consensus with Bounded Memory

“…The weakest failure detector to achieve k-set agreement in shared memory is vectorΩ k [17]. This failure detector outputs at each process a vector of size k, such that there exists some index i, such that eventually at each process the i-th index of the output is the same correct process.…”

BG Distributed Simulation Algorithm

“…Chaudhuri and Reiners [7] use BG simulation to provide a characterization of the Set Consensus Partial Order, a refinement of Herlihy's consensus-based wait-free hierarchy [10]; a formal definition of set consensus number and a study of associated respective computing power has been later provided in [9].…”

Wait-freedom with advice

“…In this section, we propose a new definition of what it means to solve a task using a failure detector and relate it to the conventional definition of [9]. Parts of our model reuse elements of [9,10,18,21].…”

“…All tasks that can be solved k-concurrently but not (k + 1)-concurrently (e.g., k-set agreement) are equivalent in the sense that they require exactly the same amount of information about failures (captured by ¬Ω k ) to be solved in EFD. Note that this characterization covers all tasks, including "colored" ones that evaded any characterization so far [13,18,1].…”