Characterizing Asynchronous Message-Passing Models Through Rounds

Message-passing models of distributed computing vary along numerous dimensions: degree of synchrony, kind of faults, number of faults... Unfortunately, the sheer number of models and their subtle distinctions hinder our ability to design a general theory of message-passing models. One way out of this conundrum restricts communication to proceed by round. A great variety of message-passing models can then be captured in the Heard-Of model, with predicates on the messages sent in a round that were received during or before this round. Characterizing a model by such a predicate then depends on how to implement rounds in the model. This is straightforward in synchronous models, thanks to the upper bound on communication delay. On the other hand, asynchrony allows unbounded message delays, which makes the implementation of rounds dependent on the specific model. We formalize the characterization of asynchronous models by Heard-Of predicates by introducing Delivered collections: the collections of all messages delivered at each round, whether late or not. Predicates on Delivered collections capture message-passing models. The question is then: which Heard-Of predicates can be generated by a given Delivered predicate? We answer it by considering all possible scheduling of deliveries and change of rounds. Strategies of processes then constrain those scheduling by specifying when processes can change rounds; those ensuring no process is ever blocked forever generate a Heard-Of collection per run, that is a Heard-Of predicate. Those strategies are then studied through a dominance relation: a dominant strategy for a Delivered predicate generates the most constrained Heard-Of predicate possible. The results of this study are the dominant Heard-Of predicates for classical asynchronous models and the existence, for every Delivered predicate, of a strategy dominating large classes of strategies.