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CHAPTER 6. VALUE SELECTION REVISED 103 This case requires that an acceptor in Q has accepted value w in some epoch ≤ f (Lemma 8.1). Since all acceptors in Q have accepted (e,v) from the monotonicity of accepted proposals (Lemmas 6 & 7), we know that acceptors in Q will return proposals from epochs ≥ e. Since the epochs e to f are limited to value v then proposals returned by acceptors in Q must be for value v. Therefore v = w. In this section, we have proven the safety of our new revision A algorithm using quorum- based value selection (Algorithm 16). We could extend this algorithm to utilise the results of revisions B and C by bypassing phase one when e = min(E) and finishing phase one if a proposal with the predecessor of e is received. Next, we will prove the correctness of the two results (Lemmas 19 & 20) utilised by Quorum-based value selection (§3). Lemma 19. If an acceptor a sends promise(f,e,w) where (e, w) = nil then no decision is reached in epochs up to f (exclusive) by the quorums containing a Proof of Lemma 19. Assume that an acceptor a sends promise(f,e,w) where (e, w) = nil. Prior to sending promise(f,e,w), the acceptor a cannot have accepted any proposal for epochs up to f since (e, w) = nil. As such no quorum containing a can have decided a proposal with an epoch up to f. Subsequently to sending promise(f,e,w), the acceptor a will not have accepted any proposal for epoch up to f as its last promised epoch will always be f or greater. As such no quorum containing a will have decided a proposal with epoch up to f. Recall Theorem 13: Theorem 13 (Safety of future proposals). If the value v is decided in epoch e and the valuewisproposedinepochf suchthatePDF Image | Distributed consensus
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