Early Decision and Stopping in Synchronous Consensus: A Predicate-Based Guided Tour

Abstract

Abstract. Consensus is the most basic agreement problem encountered in faulttolerant distributed computing: each process proposes a value and non-faulty processes must agree on the same value, which has to be one of the proposed values. While this problem is impossible to solve in asynchronous systems prone to process crash failures, it can be solved in synchronous (round-based) systems where all but one process might crash in any execution. It is well-known that (t + 1) rounds are necessary and sufficient in the worst case execution scenario for the processes to decide and stop executing, where t < n is a system parameter denoting the maximum number of allowed process crashes and n denotes the number of processes in the system. Early decision and stopping considers the case where f < t processes actually crash, f not being known by processes. It has been shown that the number of rounds that have to be executed in the worst case is then min(f + 2, t + 1). Following Castañeda, Gonczarowski and Moses (DISC 2014), the paper shows that this value is an upper bound attained only in worst execution scenarios. To this end, it investigates a sequence of three early deciding/stopping predicates P1 = Pcount, P2 = P dif and P3 = P pref0 , of increasing power, which differ in the information obtained by the processes from the actual failure, communication and data pattern. It is shown that each predicate Pi is better than the previous one Pi−1, i ∈ {2, 3}, in the sense that there are executions where Pi allows processes to reach a decision earlier than Pi−1, while Pi−1 never allows a process to decide earlier than Pi. Moreover, P3 = P pref0 is an unbeatable predicate in the sense that it cannot be strictly improved: if there is an early deciding/stopping predicate P that improves the decision time of a process with respect to P pref0 in a given execution, then there is at least one execution in which a process decides with P strictly later than with P pref0 .

Synchronous t-resilient consensus in arbitrary graphs

“…We only mention some of the most relevant papers, among a vast literature, which is covered only partially even by surveys, e.g. [8,30] and textbooks on the field, e.g. [4,26,31].…”

Parameterized Model Checking of Synchronous Distributed Algorithms by Abstraction

“…The processes use the values of the parameters n and t; the parameter f is used only by the environment. We are led to distinguish between f and t, as some of our case studies (early deciding consensus) terminate in min(f + 2, t + 1) rounds [9,11]. Full definitions of the modeling sketched in this section are found in Section 3.…”

“…Disentangling the methods from the interleaving semantics is challenging. At the same time, there is substantial literature on distributed algorithms that are not designed for interleaving semantics, namely round-based distributed algorithms [9,11,12,34,37,44,47,53]. In these algorithms, computations proceed in rounds, in which processes perform send, receive and compute transitions in lock-step.…”

“…For termination guards found in many synchronous algorithms, we provide verification conditions, which can be reused. -We do experiments on several synchronous FTDAs [9,11,37,47], some of which were not automatically verified before. Our experiments show that parameterized model checking performs better than checking fixed-size systems already for few (typically 5) processes.…”

A Simplicial Model for $KB4_n$: Epistemic Logic with Agents that May Die

“…the seminal work of Dwork and Moses [9], where a complete characterization of the number of rounds required to reach simultaneous consensus was given, in terms of common knowledge. For more recent additional references see e.g [6,12,15].…”