The concurrency hierarchy, and algorithms for unbounded concurrency

Abstract

We study wait-free computation using (read/write) shared memory under a range of assumptions on the arrival pattern of processes. We distinguish first between bounded and infinite arrival patterns, and further distinguish these models by restricting the number of arrivals minus departures, the concurrency. Under the condition that no process takes infinitely many steps without terminating, for any finite bound k > 0, we show that bounding concurrency reveals a strict hierarchy of computational models: a model in which concurrency is bounded by k + 1 is strictly weaker than the model in which concurrency is bounded by k, for all k ≱ 1. A model in which concurrency is bounded in each run, but no bound holds for all runs, is shown to be weaker than a k-bounded model for any k. The unbounded model is shown to be weaker still—in this model, finite prefixes of runs have bounded concurrency, but runs are admitted for which no finite bound holds over all prefixes. Hence, as the concurrency grows, the set of solvable problems strictly shrinks. Nevertheless, on the positive side, we demonstrate that many interesting problems (collect, snapshot, renaming) are solvable even in the infinite arrival, unbounded concurrency model.