Hundreds of impossibility results for distributed computing

Abstract. We propose a new classification for evaluating the strength of shared objects. The classification is based on finding, for each object of type o, the strongest progress condition for which it is possible to solve consensus for any number of processes, using any number of objects of type o and atomic registers. We use the strongest progress condition to associate with each object a number call the power number of that object. Objects with higher power numbers are considered stronger. Then, we define the power hierarchy which is an infinite hierarchy of objects such that the objects at level i of the hierarchy are exactly those objects with power number i. Comparing our classification with the traditional one which is based on fixing the progress condition (namely, wait-freedom) and finding the largest number of processes for which consensus is solvable, reveals interesting results. Our equivalence and extended universality results, provide a deeper understanding of the nature of the relative computational power of shared objects.

Section: Consensus Numbers and The Wait-free Hierarchymentioning

confidence: 93%

Section: Related Workmentioning

confidence: 99%

On the Computational Power of Shared Objects

Taubenfeld

2009

“…Another very simple and well-known technique for establishing impossibility results are partitioning arguments, which have been used successfully for many distributed computing problems [13]. Essentially, a partioning argument exploits the fact that one cannot guarantee agreement among those processes of a distributed system that never, neither directly nor indirectly, communicate with each other.…”

Section: Introductionmentioning

confidence: 99%

Easy Impossibility Proofs for k-Set Agreement in Message Passing Systems

Biely

Robinson

Schmid

2011

Despite of being quite similar agreement problems, consensus and general k-set agreement require surprisingly different techniques for proving the impossibility in asynchronous systems with crash failures: Rather than relatively simple bivalence arguments as in the impossibility proof for consensus (= 1-set agreement) in the presence of a single crash failure, known proofs for the impossibility of k-set agreement in systems with f k > 1 crash failures use algebraic topology or a variant of Sperner's Lemma. In this paper, we present a generic theorem for proving the impossibility of k-set agreement in various message passing settings, which is based on a simple reduction to the consensus impossibility in a certain subsystem.We demonstrate the broad applicability of our result by exploring the possibility/impossibility border of k-set agreement in several message-passing system models: (i) asynchronous systems with crash failures, (ii) partially synchronous processes with (initial) crash failures, and (iii) asynchronous systems augmented with failure detectors. In (i) and (ii), the impossibility part is just an instantiation of our main theorem, whereas the possibility of achieving k-set agreement in (ii) follows by generalizing the consensus algorithm for initial crashes by Fisher, Lynch and Patterson. In (iii), applying our technique yields the exact border for the parameter k where k-set agreement is solvable with the failure detector class (Σ k , Ω k ) 1 k n−1 of Bonnet and Raynal. Considering that Σ k was shown to be necessary for solving k-set agreement, this result yields new insights on the quest for the weakest failure detector.

“…The inability to distinguish a crashed process from a slow process makes it impossible to solve several classic problems in distributed computing in crash-prone asynchronous systems [21,22]. Efforts to circumvent this impossibility have spawned two complementary approaches.…”

Section: Introductionmentioning

confidence: 99%

Failure Detectors Encapsulate Fairness

Pike

Sastry

Welch

2010

Failure detectors have long been viewed as abstractions for the synchronism present in distributed system models. However, investigations into the exact amount of synchronism encapsulated by a given failure detector have met with limited success. The reason for this is that traditionally, models of partial synchrony are specified with respect to real time, but failure detectors do not encapsulate real time. Instead, we argue that failure detectors encapsulate the fairness in computation and communication. Fairness is a measure of the number of steps executed by one process relative either to the number of steps taken by another process or relative to the duration for which a message is in transit. We argue that failure detectors are substitutable for the fairness properties (rather than real-time properties) of partially synchronous systems. We propose four fairness-based models of partial synchrony and demonstrate that they are, in fact, the 'weakest system models' to implement the canonical failure detectors from the Chandra-Toueg hierarchy. We also propose a set of fairness-based models which encapsulate the G c parametric failure detectors which eventually and permanently suspect crashed processes, and eventually and permanently trust some fixed set of c correct processes.