From bounded to unbounded concurrency objects and back

Abstract

We consider the power of objects in the unbounded concurrency shared memory model, where there is an infinite set of processes and the number of processes active concurrently may increase without bound. By studying this model we obtain new results and observations that are relevant and meaningful to the standard bounded concurrency model.First we resolve an open problem from 2006 and provide, contrary to what was conjectured, an unbounded concurrency wait-free implementation of a swap object from 2-consensus objects. This construction resolves another puzzle that has eluded us for a long time, that of considerably simplifying a 16 year old complicated bounded concurrency swap construction.A further insight to the traditional bounded concurrency model that we obtain by studying the unbounded concurrency model, is a refinement of the top level of the waitfree hierarchy, the class of infinite-consensus number objects. First we resolve an open question of Merritt and Taubenfeld from 2003, showing that having n-consensus objects for all n does not imply consensus under unbounded concurrency. I.e., consensus alone, treated as a black box, cannot be "boosted" in this way. We continue to show an infinitenumber consensus object that while able to perform consensus for any n-bounded concurrency (n unknown in advance) cannot solve consensus in the face of unbounded concurrency. This divides the infinite-consensus class of objects into two, those that can solve consensus for unbounded concurrency, and those that cannot.

Notes on Theory of Distributed Systems

“…To show Algorithm 22.5 works, we need the following technical lemma, which, among other things, implies that node 1 − 2 depth is always available to be captured by the process at depth depth. This is essentially just a restatement of Lemma 1 from [AMW11].…”

“…The construction of the generalized fetch-and-add is pretty nasty, so we'll concentrate on the implementation of swap objects. We will also skip the swap implementation in [AWW93], and instead describe, in § §22.3 and 22.4, a simpler (though possibly less efficient) algorithm from a later paper by Afek, Morrison, and Wertheim [AMW11]. Before we do this, we'll start with some easier results from the older paper, including an implementation of n-process test-and-set from 2-process consensus.…”

“…Algorithm 22.5: Wait-free swap from test-and-set [AMW11] capture any y in [1 − 2 q , 1) before I write 1 − 2 q to accessed. But this means that nobody can block me from capturing 1 − 2 q , because processes can only block values smaller than the one they already captured.…”

Extending the Wait-free Hierarchy to Multi-Threaded Systems

“…Similarly to Proposition 4, the proof of Proposition 6 builds a scheduler that builds a Π ′ -partitioned execution, keeping track of a subset Π ′ of processes that have never communicated with each other, and in which more and more shared objects are covered (Definition 7 adapts the notion of coverage to take iterator stacks into account). The major difficulty is that iterator stacks cannot be overwritten by a finite number of processes, and the valency-based proof introduced in [Afek et al 2011] cannot be adapted to a setting where binary consensus objects can be used in a critical configuration. Lemma 10 allows the scheduler to introduce a flow of newly arrived processes that, by covering, reading or writing all iterator stacks, prevents any chosen process trying to access an iterator stacks from learning any valuable information about the existence of other processes.…”

Wait-Free Universality of Consensus in the Infinite Arrival Model

“…The aim of this paper is to extend universality of consensus to the infinite arrival model. Solutions to the consensus problem have already been investigated for the infinite arrival model [2,5,13] and consensus has been used as a base for reasoning about computability in this model [1]. The question is thus "is it possible to build a universal wait-free linearizable construction based on consensus objects and read/write atomic registers?"…”