On Deterministic Linearizable Set Agreement Objects

A recent work showed that, for all n and k, there is a linearizable (n,k)-set agreement object O_L that is equivalent to the (n,k)-set agreement task [David Yu Cheng Chan et al., 2017]: given O_L, it is possible to solve the (n,k)-set agreement task, and given any algorithm that solves the (n,k)-set agreement task (and registers), it is possible to implement O_L. This linearizable object O_L, however, is not deterministic. It turns out that there is also a deterministic (n,k)-set agreement object O_D that is equivalent to the (n,k)-set agreement task, but this deterministic object O_D is not linearizable. This raises the question whether there exists a deterministic and linearizable (n,k)-set agreement object that is equivalent to the (n,k)-set agreement task. Here we show that in general the answer is no: specifically, we prove that for all n ? 4, every deterministic linearizable (n,2)-set agreement object is strictly stronger than the (n,2)-set agreement task. We prove this by showing that, for all n ? 4, every deterministic and linearizable (n,2)-set agreement object (together with registers) can be used to solve 2-consensus, whereas it is known that the (n,2)-set agreement task cannot do so. For a natural subset of (n,2)-set agreement objects, we prove that this result holds even for n = 3

A separation of (n -1)-consensus and n-consensus in read-write shared-memory systems

A fundamental research theme in distributed computing is the comparison of systems in terms of their ability to solve basic problems such as consensus that cannot be solved in completely asynchronous systems. In particular, in a seminal work [14], Herlihy compares shared-memory systems in terms of the shared objects that they have: he proved that there are shared objects that are powerful enough to solve consensus among n processes, but are too weak to solve consensus among n + 1 processes; such objects are placed at level n of a wait-free hierarchy. The importance of this hierarchy comes from Herlihy's universality result: intuitively, every object at level n of this hierarchy can be used to implement any object shared by n processes in a wait-free manner. As in [14], we compare shared-memory systems with respect to their ability to solve consensus among n processes. But instead of comparing systems defined by the shared objects that they have, we compare readwrite systems defined by the process schedules that they allow. These systems capture a large set of systems, e.g., systems whose synchrony ranges from fully synchronous to completely asynchronous, several systems with failure detectors, and "obstruction-free" systems. In this paper, we consider read-write systems defined in terms of process schedules, and investigate the following natural question: For every n, is there a system of n processes that is strong enough to solve consensus among every subset of n -1 processes in the system, but not enough to solve consensus among all the n processes of the system? We show that the answer to the above question is "yes", and so these systems can be classified into hierarchy akin to Herlihy's hierarchy

Correctness Proof of Ben-Or's Randomized Consensus Algorithm

We present a correctness proof for Ben-Or's Randomized Consensus Algorithm for the case in which processes can fail by crashing, and a majority of processes is correct. This is the first time that the proof of Ben-Or's algorithm appears for this case. The proof has been extracted from [AT96]: it is a simplification of the correctness proof of a more complex consensus algorithm that involves both randomization and failure detection

On the Number of Objects with Distinct Power and the Linearizability of Set Agreement Objects

We first prove that there are uncountably many objects with distinct computational powers. More precisely, we show that there is an uncountable set of objects such that for any two of them, at least one cannot be implemented from the other (and registers) in a wait-free manner. We then strengthen this result by showing that there are uncountably many linearizable objects with distinct computational powers. To do so, we prove that for all positive integers n and k, there is a linearizable object that is computationally equivalent to the k-set agreement task among n processes. To the best of our knowledge, these are the first linearizable objects proven to be computationally equivalent to set agreement tasks

Bounded Disagreement

A well-known generalization of the consensus problem, namely, set agreement (SA), limits the number of distinct decision values that processes decide. In some settings, it may be more important to limit the number of "disagreers". Thus, we introduce another natural generalization of the consensus problem, namely, bounded disagreement (BD), which limits the number of processes that decide differently from the plurality. More precisely, in a system with n processes, the (n, l)-BD task has the following requirement: there is a value v such that at most l processes (the disagreers) decide a value other than v. Despite their apparent similarities, the results described below show that bounded disagreement, consensus, and set agreement are in fact fundamentally different problems. We investigate the relationship between bounded disagreement, consensus, and set agreement. In particular, we determine the consensus number for every instance of the BD task. We also determine values of n, l, m, and k such that the (n, l)-BD task can solve the (m, k)-SA task (where m processes can decide at most k distinct values). Using our results and a previously known impossibility result for set agreement, we prove that for all n >= 2, there is a BD task (and a corresponding BD object) that has consensus number n but can not be solved using n-consensus and registers. Prior to our paper, the only objects known to have this unusual characteristic for n >= 2 (which shows that the consensus number of an object is not sufficient to fully capture its power) were artificial objects crafted solely for the purpose of exhibiting this behaviour