Stabilizing Agreement is Impossible in Delayed Message Passing Models

Most distributed computing research has focused on terminating problems like consensus and similar agreement problems. Non-terminating problems have been studied exhaustively in the context of self-stabilizing distributed algorithms, however, which may start from arbitrary initial states and can tolerate arbitrary transient faults. Somehow in-between is the stabilizing consensus problem, where the processes start from a well-defined initial state but do not need to decide irrevocably and need to agree on a common value only eventually. Charron-Bost and Moran studied stabilizing consensus in synchronous dynamic networks controlled by a message adversary. They introduced the simple and elegant class of min-max algorithms, which allow to solve stabilizing consensus under every message adversary that (i) allows at least one process to reach all other processes infinitely often, and (ii) does so within a bounded (but unknown) number of rounds. Moreover, the authors proved that (i) is a necessary condition. The question whether (i) is also sufficient, i.e., whether (ii) is also necessary, was left open. We answer this question by proving that stabilizing consensus is impossible if (ii) is dropped, i.e., even if some process reaches all other processes infinitely often but only within finite time. We accomplish this by introducing a novel class of arbitrarily delayed message adversaries, which also allows us to establish a connection between terminating task solvability under some message adversary to stabilizing task solvability under the corresponding arbitrarily delayed message adversary. Finally, we outline how to extend this relation to terminating task solvability in asynchronous message passing with guaranteed broadcasts, which highlights the asynchronous characteristics induced by arbitrary delays.