Unreliable Failure Detectors for Reliable Distributed Systems

252 T. D. CHANDRAAND S. TOUEGpropose(k, msgSetk). From our definition of k, q is correct and msgSetkcontains m. Thus, all correct processes, including p, A-deliver m—a contradictionthat concludes the proof. ❑LEMMA 7.1.5 (UNIFORM INTEGRITY). For any nzessuge nz, each process A-delivers m at most once, and only if m was previously A-broadcast by sender(m).PROOF. Suppose a process p A-delivers m. After p A-delivers m, itinserts min A _deliveredP. From the algorithm, it is clear that p cannot A-deliver m again.From the algorithm, p executed decide(k, msgSetk) for some k and somemsgSefk that contains m. By uniform validity of Consensus, some process q musthave executed propose(k, msgSetk). So q previously R-delivered all the messagesin msgSetk, including m. By the uniform integrity property of ReliableBroadcast, process sender(m) R-broadcast m. So, sender(m) A-broadcast m. ❑THEOREM 7.1.6. Consider any system (synchronous or asynchronous) subject tocrash failures and where Reliable Broadcast can be implemented. The algorithm inFigure 7 transfomrs any algorithm for Consensus into an Atomic Broadcastalgorithm.PROOF. Immediate from Lemmata 7.1.3, 7.1.4, and 7.1.5. •!Since Reliable Broadcast can be implemented in asynchronous systems withcrash failures (Section 4), the above theorem shows that Atomic Broadcast isreducible to Consensus in such systems. As we showed earlier, the converse isalso true. Thus:zlCOROLLARY7.1.7. Consensus and Atomic Broadcast are equivalent in asynchronoussystems,This equivalence immediately implies that our results regarding Consensus (inparticular Corollaries 6.1.9 and 6.2.6, and Theorem 6.3.1) also hold for AtomicBroadcast:COROLLARY7.1.8. Atomic Broadcast is solvable u.ring W in a~nchronoussystems with f < n, and using OW in asynchronous systems with f < rn/21.COROLLARY7.1.9. OWO b the weakest failure detector for solving AtomicBroadcast in asynchronous systems with f < rn/21.COROLLARY 7.1.10. Atomic Broadcast cannot be solved using 09 in asynchronoussystems with f z [n/21.Furthermore, Theorem 7.1.6 shows that by “plugging in” any randomuedConsensus algorithm (such as the ones in Chor and Dwork [1989]) into thealgorithm of Figure 7, we automatically get a randomized algorithm for AtomicBroadcast in asynchronous systems.COROLLARY7.1.11. Atomic Broadcast can be solved by randomized algorithmsin asynchronous systems with f < rn/21.21 ~] ~h~ ~e~ult~ stated hencefofih in this section are for systems with crash failures.