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CHAPTER 4. QUORUM INTERSECTION REVISED 81 As before, we begin by revising our definition of decided: Definition 10. A proposal (e,v) is decided if the proposal (e,v) has been accepted by a phase two quorum of acceptors for epoch e. Our next result observes that we can further weaken the quorum intersection requirement. We require only that a phase one quorum of the epoch e (Qe1) intersects with the phase two quorums (Qf2) for all smaller epochs, {f ∈ E|f < e}1. There is no requirement that the acceptors for the phase one and phase two quorums for a given epoch intersect. Likewise, there is no requirement that a phase one quorum of the epoch must intersect with the phase two quorums for all greater epochs. This newly revised quorums intersection requirement, referred to as the revision B quorum intersection requirement, can be specified as follows for each epoch e: ∀Q ∈ Qe1, ∀f ∈ E : f < e =⇒ ∀Q′ ∈ Qf2 : Q ∩ Q′ ̸= ∅ (4.6) 4.2.1 Algorithm Algorithm 14 gives the revised generalised pseudo-code for Classic Paxos. Only the proposer algorithm is provided here as the acceptor algorithm is unchanged from Algorithm 4. Note that it is now possible for a phase one quorum to be empty (as we will discuss later) thus we can add the option to skip phase one in this case. 4.2.2 Safety Similar to the case of revision A (Equation 4.4), the safety of this result is derived from the observation that the proof of safety for Classic Paxos does not use the full strength of the assumptions made regarding quorum intersection. 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. As before, we begin by redefining Properties 2 & 3. All other properties remain unchanged. 1Or equivalently, that a phase two quorum of an epoch e intersects with the phase one quorums from all greater epochs.PDF Image | Distributed consensus
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