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CHAPTER 7. EPOCHS REVISED 123 Property 4. Proposers must choose a value to propose according to the value selection rules. If no previously accepted proposals were returned with promises then any value can be chosen. If one or more previously accepted proposals were returned then the value associated with the highest epoch is chosen. However, we will add the following three additional properties for future use: Property 17. For each propose message received by an acceptor where the epoch received is the same as the last accepted epoch, the message is processed by the acceptor only if the proposed value is the same as the last accepted value. Property 18. Proposers only propose a value after receiving promises from sufficient acceptors such that at most one value may have been decided. Property 19. Proposers must choose a value to propose in epoch e according to the value selection rules. If Vdec is an empty set then any value can be chosen. Otherwise if Vdec is a singleton then its only value is chosen. From property 17 it follows that: Lemma 26. An acceptor will not accept more than one proposal with a given epoch. If an acceptor accepts (e, v) and (e, w) for any epoch e ∈ E then v = w. Proof of Lemma 26. Assume that an acceptor has accepted (e, v) then (e, w). From Prop- erties 10, 6 & 9, the last accepted proposal must be (e, v) when (e, w) is accepted. From Property 17, then v = w. We can therefore show that: Lemma 27. If the value v is decided in epoch e then no other value w where v ̸= w will also be decided in e. Proof of lemma 27. Assume the proposal (e, v) has been decided therefore a phase two of acceptors Q ∈ Q2 have accepted (e, v). Likewise for w to be decided, a phase two quorum of acceptors Q′ ∈ Q2 must have accepted (e,w). As any two phase two quorums for a given epoch intersect, then at least one acceptor must have accepted both proposals. From lemma 26 then v = w, so no other value can be accepted. We begin be revising our proof of Corollary 12.1. Corollary 12.1 (Base case for safety of future proposals). If the value v is decided in epoch e and the value w is proposed succ(e) then v = w.PDF Image | Distributed consensus
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