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76 4.1. QUORUM INTERSECTION ACROSS PHASES Our first finding is that it is only necessary for phase one quorums (Q1) and phase two quorums (Q2) to intersect. There is no need to require that phase one quorums intersect with each other nor that phase two quorums intersect with each other. Since no intersection is required within phases, quorums within each phase of Classic Paxos can be disjoint. We will refer to this generalisation of Classic Paxos as Paxos revision A. In the literature, we referred to this as Flexible Paxos (FPaxos). Formally, the revision A quorum intersection requirement can be stated as: ∀Q1 ∈Q1,∀Q2 ∈Q2 :Q1 ∩Q2 ̸=∅ (4.4) 4.1.1 Algorithm Algorithm 13 gives the generalised pseudo-code for Classic Paxos. Only the proposer algorithm is provided here as the acceptor algorithm is unchanged from Algorithm 4. We could configure the algorithm with the quorums sets Q1 and Q2, or we can provide one of the quorums sets and calculate the other as needed. The latter is the approach taken in Algorithm 13. 4.1.2 Safety We will now consider why it is safe to relax the quorum intersection requirement by examining how the intersection of quorums was utilised in the earlier safety proof for Classic Paxos (§2.6). Recall the following properties (originally defined §2.4): Property 2. Proposers only propose a value after receiving promises from ⌊na/2⌋ + 1 acceptors. Property 3. Proposers only return a value after receiving accepts from ⌊na/2⌋ + 1 accep- tors. We will now replace them with the following properties for Paxos revision A. All other properties remain unchanged. Property 11. Proposers only propose a value after receiving promises from a phase one quorum of acceptors, Q ∈ Q1. Property 12. Proposers only return a value after receiving accepts from a phase two quorum of acceptors, Q ∈ Q2.PDF Image | Distributed consensus
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