Knowledge and common knowledge in a Byzantine environment: Crash failures

By analyzing the states of knowledge that the processors attain in an unreliable system of a simple type, we capture some of the basic underlying structure of such systems. In particular, we study what facts become common knowledge at various points in the execution of protocols in an unreliable system. This characterizes the simultaneous actions that can be carried out in such systems. For example, we obtain a complete characterization of the number of rounds required to reach simultaneous Byzantine agreement, given the pattern in which failures occur. From this we derive a new protocol for this problem that is optimal in all runs, rather than just always matching the worst-case lower bound. In some cases this protocol attains simultaneous Byzantine agreement in as few as two rounds. We also present a nontrivial simultaneous agreement problem called bivalent agreement for which there is a protocol that always halts in two rounds. Our analysis applies to simultaneous actions in general, and not just to Byzantine agreement. The lower bound proofs presented here generalize and simplify the previously known proofs.