Classifications of Asynchronous Shared Memory Objects

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 consensus number of an object [24]. 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 object cannot be wait-free implemented by instances of the other and registers) [2, 42]. A more recent candidate for such a characterization is the set agreement power of an object [15]. In this thesis, we show that this characterization is not precise even for the case of deterministic objects: there are pairs of deterministic objects with the same set agreement power that are not equivalent. Next, we show that there are uncountably many objects with distinct set agreement power. Consequently, a precise characterization of an object’s ability to implement other objects in a wait-free manner must classify objects into uncountably many cells. This suggests that there is no simple and precise characterization of objects, since the cells of an uncountable classification cannot be labeled using simple integers (as in Herlihy’s hierarchy) or even finite sequences of integers. Finally, we introduce a generalization of consensus called bounded disagreement, which differs from set agreement in that it restricts the number of disagreeing processes rather than the number of disagreeing values. More precisely, the l-bounded disagreement task for n processes has the following requirement: there is a value v such that at most l processes (the disagreers) decide values that are not v. We then show that for every integer n ≥ 2, there is a bounded disagreement object that has consensus number n, but is not equivalent to the n-consensus object. Prior to this work, the only objects known to have this unusual characteristic for n ≥ 2 (which shows that the characterization of objects via consensus numbers is not precise) were artificial objects crafted solely for the purpose of exhibiting this behaviour [2, 42].Ph.D