Revisionist Simulations: A New Approach to Proving Space Lower Bounds

Determining the number of registers required for solving x-obstruction-free (or randomized wait-free) k-set agreement for x ≤ k is an open problem that highlights important gaps in our understanding of the space complexity of synchronization. In x-obstruction-free protocols, processes are required to return in executions where at most x processes take steps. The best known upper bound on the number of registers needed to solve this problem among n>k processes is n-k+x registers. No general lower bound better than 2 was known. We prove that any x-obstruction-free protocol solving k-set agreement among n > k processes must use n-x/k+1-x \rfloor + 1 or more registers. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free k-set agreement. In particular, we show that, if a protocol uses fewer registers, then it is possible for k+1 processes to simulate the protocol and deterministically solve k-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. We also 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 bound of n-1/k + 1 for the obstruction-free case (i.e., x = 1) also holds for randomized wait-free protocols. In particular, we get a tight lower bound of exactly n registers for solving obstruction-free and randomized wait-free consensus.