Lower Bounds for Achieving Synchronous Consensus with Orderly Crash Failure

The time lower bounds for both uniform consensus and early stopping consensus have already been proved in synchronous distributed systems with crash failures, but both have not been proved with orderly crash failures and remain as open problems. This paper uses a bivalency argument proof to solve the two open problems. The first open problem is how to prove the lower bound, min (t + 1, f + 2) rounds, for early-stopping synchronous consensus with orderly crash failures, where t < n - 1 and 0 les f les t -1. The second problem is to prove that the lower bound for early deciding uniform consensus with orderly crash failures is (f + 2)-rounds where 0 les f les t -2, and (f + 1)-rounds when f = t and t - 1.