On the Classification of Deterministic Objects via Set Agreement Power

Since the early days of the shared memory model for distributed computing, researchers have sought a simple and precise characterization of an object's ability to implement other objects in a wait-free manner. The first candidate for such a characterization was the object's consensus number [14]. But a characterization based on consensus numbers is not precise: there are pairs of objects with the same consensus number that are not equivalent (i.e., one cannot be implemented by instances of the other and registers). This was first shown for non-deterministic objects [24] and much later for deterministic objects as well [2]. A more recent candidate for such a characterization is the object's set agreement power [10]. In PODC 2017, it was shown that this characterization is also not precise: there are two objects with the same set agreement power that are not equivalent [6]. One of these two objects, however, is non-deterministic. So this left open one remaining final question: when restricted to deterministic objects, does the set agreement power of an object fully characterize its ability to implement other objects? More precisely: are any two deterministic objects with the same set agreement power equivalent? In this paper, we show the answer is again no. In fact, we prove the following stronger result: every level n ≥ 2 of Herlihy's consensus hierarchy [14] contains two deterministic objects with the same set agreement power that are not equivalent (i.e., one of these two objects cannot be implemented using instances of the other and registers). We also leverage the above result and a result in [12] to show that in any system with n > 2 processes, there is a deterministic wait-free object that is not equivalent to any wait-free task.