Distributed consensus

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Distributed consensus ( distributed-consensus )

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CHAPTER 7. EPOCHS REVISED 125 Consider the case that Vdec = {w}. For all quorums including Q, D[Q] = no or D[Q] = w. As we have already shown it cannot be the case that D[Q] = no thus D[Q] = w and therefore for some acceptor ∃a ∈ Q : R[a] = (h,w). This acceptor must have first accepted (e,v) thus h ≥ e. If h = e thenv=w(Lemma26).Otherwise,ife f. From the symmetry oftheorem14,wecanassumeef byswappingeandf. For a value to be decided, it must first be proposed, therefore a stronger theorem is theorem 13. 7.3.4 Progress Earlier, we claimed that the strengthened quorum intersection requirement in Equation 7.2 is always sufficient to make progress. Now we will examine this claim. Lemma 28. After a proposer in epoch e has received sufficient promises to satisfy Equa- tion 7.2, possibleValues always returns an empty or singleton set. Proof of lemma 28. Consider a proposer in epoch e who is calling possibleValues after receiving sufficient promises to satisfy Equation 7.2. Assume that possibleValues returns a set of two or more values such as {v, v′, ...} where v ̸= v′. It is therefore the case that there are two quorums Q, Q′ ∈ Q2 such that D[Q] = v and D[Q′] = v′. Thisrequiresthat∀a∈Q:R[a]=no∨R[a]=(,v)and∀a∈Q′ :R[a]=no∨R[a]= ( ,v′).

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