An Asynchronous Graph Characterization of Byzantine Firing Rebels

Byzantine agreement tasks have been studied extensively through many diverse frameworks ranging from epistemic modal logic to combinatorial, topological and even game theoretical approaches. Among byzantine agreement tasks, firing rebels with relay is of particular interest, since it is a basic primitive necessary for solving more complex tasks such as Consistent Broadcast. The epistemic logic approach has yielded both necessary conditions and sufficient knowledge conditions for solving firing rebels with relay. However, these conditions are stated in terms of knowledge and in principle do not explore the conditions on the communication structure which is often assumed to be complete. That is, any process is assumed to be capable of communicating with any other process at any time. In this paper, we characterize byzantine firing rebels solvability with and without relay in terms of the communication structure of the system. We define a relation between asynchronous message schedules and directed graph sequences, which we call network abstractions. This allows us to capture the message relay capabilities of the system into a combinatorial object. Although there are some similarities between network abstractions and causal cones, there is a fundamental difference. Namely, causal cones allow only to look at events in the past, while network abstractions allow us to reason about future possibilities. Thus enabling us to reason about liveness properties, which can not be expressed by looking only at past events. Furthermore, we formalize our characterization by using a temporal epistemic logic for byzantine systems. Such formulation constitutes the necessary and sufficient a-priori knowledge regarding network connectivity for solving firing rebels with relay.