Revisionist Simulations: A New Approach to Proving Space Lower Bounds

Abstract

SIAM Journal on Computing, Volume 53, Issue 4, Page 1039-1084, August 2024.
Abstract. Determining the number of registers required for solving obstruction-free (or randomized wait-free) [math]-set agreement is an open problem that highlights important gaps in our understanding of the space complexity of synchronization. The best known upper bound on the number of registers needed to solve this problem among [math] processes is [math] registers. No general lower bound better than 2 was known. We prove that any obstruction-free protocol solving [math]-set agreement among [math] processes must use at least [math] registers. In particular, we get a tight lower bound of exactly [math] registers for solving obstruction-free and randomized wait-free consensus. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free [math]-set agreement. In particular, we show that if an obstruction-free protocol for [math]-set agreement uses fewer registers, then it is possible for [math] processes to simulate the protocol and deterministically solve [math]-set agreement in a wait-free manner, which is impossible. An important aspect of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce an augmented snapshot object, which facilitates this. More generally, our simulation applies to the broad class of colorless tasks. We can use it to prove, for example, a lower bound on the number of registers needed to solve obstruction-free [math]-approximate agreement, which matches the best known upper bound to within a factor of 2 when [math] is sufficiently small. No general lower bound for this problem was known. Finally, we prove that any lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo-termination. Hence, our lower bounds for obstruction-free protocols also hold for randomized wait-free protocols.