## Understanding Paxos and Distributed Consensus

```(minor wording correction and more complaining added 10/2/2016,
minor edits 10/5/2016)```

Multi-proposer Paxos is a very clever and notoriously slippery algorithm for obtaining distributed consensus. In this note I try to explain it clearly and provide a correctness proof that gives some intuition why it works – to the extent that it does work. I am specifically concerned with Multi-Proposer Paxos, the first algorithm discussed in “Paxos Made Simple”.  What is often called “Paxos” involves single Proposer variants which are much simpler and less interesting.

I think this is right – let me know if you spot an error.

## Rules for how Paxos works

There is a finite set of processes or network sites, some of which are Proposers and some Acceptors (the sets can intersect). Each proposer has a unique id, confusingly called a sequence number. A proposal is a pair consisting of a proposal value and the sequence number of the Proposer. The goal is to ensure that if two Proposers believe that they have convinced the Acceptors to come to a consensus on a value, they must both agree on the same value, even though they may disagree on sequence number. The most clever part of Paxos is the observation that since we don’t care which value wins, even though we do care that some unique value wins, we can force Proposers to inherit values of the most likely previous proposal.

1. Proposers can ask Acceptors to approve sequence numbers and to accept proposals which include a value and the Proposer’s sequence number. Acceptors do not have to approve or accept but are limited to approving and accepting what Proposers send them.
2. When an Acceptor approves a sequence number it:
1. Promises to not approve any smaller sequence numbers
2. Promises to not accept any proposals with smaller sequence numbers
3. Returns to the Proposer the proposal with the highest sequence number it has already accepted, if any.
3. The Proposer cannot send any proposals or select a value for a proposal until it gets approval for its sequence number from a majority of Acceptors.
4. Once the Proposer has approval from a majority of Acceptors it must select the value of the proposal with the highest sequence number sent to it during the approval phase (the inherited proposal). If the approval phase did not turn up any accepted proposals, the Proposer can pick any value. In this case the Proposer “originates” the value.
5. Once the value is selected, the Proposer can never change the value and can only propose the pair of that value and its sequence number – unless it increases its sequence number, abandons the proposal and value, and starts over.
6. The choice of a new sequence number must preserve the property that each sequence number belongs to only one Proposer, ever.

(see the  code for what this looks like in a simulation)

## Why it works intuition

The first thing to see is that individual Acceptors are forced to order their approvals and accepts by sequence number. If an Acceptor has approved j and accepted (i,v) and j>i then we know that it must have approved j after it accepted (i,v). The sequence of operations for that Acceptor must act like this: