Distributed consensus, the ability to reach agreement in the face of failures, is a fundamental primitive for constructing reliable distributed systems. The Paxos algorithm is synonymous with consensus and widely utilized in production. Paxos uses two phases: phase one and phase two, each requiring a quorum of acceptors, to reach consensus during a round of the protocol. Traditionally, Paxos requires that all quorums, regardless of phase or round, intersect and majorities are often used for this purpose. Flexible Paxos proved that it is only necessary for phase one quorum of a given round to intersect with the phase two quorums of all previous rounds.

In this paper, we re-examine how Paxos approaches the problem of consensus. We look again at quorum intersection in Flexible Paxos and observe that quorum intersection can be safely weakened further. Most notably, we observe that if a proposer learns that a value was proposed in some previous round then its phase one no longer needs to intersect with the phase two quorums from that round or from any previous rounds. Furthermore, in order to provide an intuitive explanation of our results, we propose a novel abstraction for reasoning about Paxos which utilizes write-once registers.