A Complexity-Based Hierarchy for Multiprocessor Synchronization

Abstract

For many years, Herlihy's elegant computability-based Consensus Hierarchy has been our best explanation of the relative power of various types of multiprocessor synchronization objects when used in deterministic algorithms. However, key to this hierarchy is treating synchronization instructions as distinct objects, an approach that is far from the real-world, where multiprocessor programs apply synchronization instructions to collections of arbitrary memory locations. We were surprised to realize that, when considering instructions applied to memory locations, the computability based hierarchy collapses. This leaves open the question of how to better capture the power of various synchronization instructions.In this paper, we provide an approach to answering this question. We present a hierarchy of synchronization instructions, classified by the space complexity necessary to solve consensus in an obstructionfree manner using these instructions. Our hierarchy provides a classification of combinations of known instructions that seems to fit with our intuition of how useful some are in practice, while questioning the effectiveness of others. In particular, we prove an essentially tight characterization of the power of buffered read and write instructions. Interestingly, we show a similar result for multi-location atomic assignments.

Space Complexity of Fault-Tolerant Register Emulations

“…The proof of our main result (see Lemma 1) further extends the adversarial framework of [37] to exploit the notion of register covering (originally due to [10]) extended to fault-prone base registers as in [2]. Covering arguments have been successfully applied to proving numerous space lower bound results in the literature (see [6] for a survey) including the recent tight bounds for obstruction-free consensus [20,39], which are at the heart of the space hierarchy of [17].…”

“…In addition, since atomicity usually requires readers to write, it is interesting to investigate whether the space complexity (assuming read/write registers) in this case also linearly depends on the number of readers. Our results suggest a new classification of the data types based on space complexity of faulttolerant emulations built from base objects of these types, which is fundamentally different from those established by [23] and [17]. A promising future direction is to extend this classification with additional types (e.g., multiple assignment), and potentially generalize it into a full-fledged hierarchy of its own.…”

Extending the Wait-free Hierarchy to Multi-Threaded Systems

“…Proposition 4 below shows that neither binary consensus nor window registers are universal in the finite arrival model when infinite memory allocation is impossible. The proof has the same flavor as the proofs in [Ellen et al 2016], but simplified as we are only interested in decidability whereas their bounds need to be tight. More precisely, the proof of Proposition 4 builds a scheduler that keeps track of a subset Π ′ of processes that have never communicated with each other because they always propose the same values in binary consensus objects, and the values they write in window registers are overwritten.…”

A Simple Object that Spans the Whole Consensus Hierarchy

“…It follows that the schedule op j σ j (executed previously from Σ) can also be executed from Σ i . The first k operations of this schedule are a write operation on KREG issued by each process different 4 The intuition that underlies this case is the following. While pi can be the first process that writes a value (say 0) in KREG (thereby producing a 0-valent configuration) and then pauses for an arbitrarily long period, it is possible that the next process writes 1, and the (k − 1) other processes write also a value, whose net effect is the elimination of the value written by pi from the current window.…”