Life Beyond Set Agreement

The set agreement power of a shared object O describes O's ability to solve set agreement problems: it is the sequence (n_1, n_2, ..., n_k, ...) such that, for every k >= 1, using O and registers one can solve the k-set agreement problem among at most n_k processes. It has been shown that the ability of an object O to implement other objects is not fully characterized by its consensus number the first component of its set agreement power) [1, 3, 14]. This raises the following natural question: is the ability of an object O to implement other objects fully characterized by its set agreement power? We prove that the answer is no: every level n >= 2 of Herlihy's consensus hierarchy has two objects that have the same set agreement power but are not equivalent, i.e., at least one cannot implement the other. We also show that every level n >= 2 of the consensus hierarchy contains a deterministic object O_n with some set agreement power (n_1, n_2, ..., n_k, ...) such that being able to solve the k-set agreement problems among n_k processes, for all k >= 1, is not enough to implement O_n.