Making Paxos face facts

Lamport’s  “Paxos Made Simple” paper is notoriously hard to understand but at least part of the difficulty is that the algorithm  changes radically in the middle of the presentation.  The first part of the paper presents a subtle (maybe too subtle) method to permit multiple processes or network sites to agree on a consensus value.  The second part of the paper switches to a second, much simpler algorithm without much notice.

The paper begins with a problem statement:

Assume a collection of processes that can propose values. A consensus algorithm ensures that a single one among the proposed values is chosen. If no value is proposed, then no value should be chosen. If a value has been chosen, then processes should be able to learn the chosen value. The safety requirements for consensus are:

  • Only a value that has been proposed may be chosen,
  • Only a single value is chosen, and
  • A process never learns that a value has been chosen unless it actually has been

The Paxos algorithm that is  presented first is liable to what we used to call livelock even in the absence of failure:

It’s easy to construct a scenario in which two proposers each keep issuing a sequence of proposals with increasing numbers, none of which are ever chosen.

It’s argued that this original Paxos works in the sense that it never can reach an inconsistent state (it is “safe” ), but the livelock scenario means it can easily fail to progress even without process failure or loss of messages. To avoid this scenario, on page seven of an eleven page paper, Paxos is redefined to restrict the set of  proposers to a single member – a distinguished proposer.

To guarantee progress, a distinguished proposer must be selected as the only one to try issuing proposals

Then, on page nine, when going over how Paxos can be used to build replicated state machines, Lamport writes:

In normal operation, a single server is elected to be the leader, which acts as the distinguished proposer (the only one that tries to issue proposals) in all instances of the consensus algorithm

Since much of the complexity of the first part of the paper involves describing how Paxos safely resolves contention between competing proposers, the modification is far from minor. It’s unfortunate that both the multi-proposer and single proposer algorithm are called by the same name which seems to cause confusion in the literature and certainly obscures the presentation in “Paxos Made Simple“. For example, the “Paxos Made Live”   paper appears to discuss an implementation based on the first (the multi-proposer) Paxos algorithm but then appears to revert to the second method via a mechanism called “master leases”).

In any case, the single proposer problem is much simpler than the original problem.

 

A brute force commit solution to the single proposer problem.

What, exactly do the bolded words in the problem statement mean?  “Chosen” and “Learn” are the two hard ones. “Proposed” is pretty clear: a process sends messages to all the other processes that says ” I , process A, propose value V with supporting documentation x“.

A proposer sends a proposed value to a set of acceptors. An acceptor may accept the proposed value. The value is chosen when a large enough set of acceptors have accepted it.

Proposal is a simple action: the proposer sends a proposal message.

There are the usual assumptions: messages may be lost or duplicated or delivered out of order, but are neither corrupted or spurious. That is: if a process receives a message from a second process, the second process must have previously transmitted that message.   A process can propose a value, but that proposal may never arrive at any of the other process or it may arrive at all of them or some of them. It makes sense that an accept operation is the transmit of an accept message by some process  that received the proposal message.  Presumably then, the value is chosen when the original proposer receives accept messages from a large enough set of the Acceptor processes  – a majority of the processes. Learning is also simple:

To learn that a value has been chosen, a learner must find out that a proposal has been accepted by a majority of acceptors.

Suppose that there is a distinguished proposer process D. Suppose that process sends value to all Acceptors and marks the value as “chosen” if it receives an accept response from a majority of Acceptors. Acceptors only send accept messages to the distinguished proposer if they receive a proposal from the distinguished proposer.  Let Learners ask the distinguished proposer for chosen values. To make the learning process more efficient, the distinguished proposer can notify Acceptors that the value has been chosen and they can answer Learners too. All properties are now satisfied:

  • Only a value that has been proposed may be chosen: only D proposes so all it has to do is to select a single value.
  • Only a single value is chosen: follows from the first.
  • A process never learns that a value has been chosen unless it actually has been: the process D knows reliably and can inform users and Acceptors that have received notification from D can also inform Learners reliably.

If there are no failures, there is nothing else to discuss. If processes can fail, there is still not much to discuss. As long as the distinguished proposer doesn’t fail, everything works. If the distinguished proposer and some  of the Acceptors fail before any Acceptor has accepted, a poll of Acceptors will reveal no accepted value – and we know that no Learner has been falsely told there is some consensus value. If some Learner was told about a chosen value and the leader and some minority of Acceptors fail,  since a majority of Acceptors must have accepted and a minority have failed, there is at least one Acceptor that knows the accepted value and it can be copied to all surviving Acceptors. There cannot be two different accepted values. If  no Learner was told about an accepted value, either no value was chosen or one was and nobody was informed. In either case, we can just copy an accepted value if any of the Acceptors has one, or start from a blank slate otherwise. No harm no foul.

Notice, we have not had to worry about reliable store. All we need is the absence of spurious or corrupted messages and the survival of at least 1/2 of the Acceptors. If we need a sequence of values, the distinguished proposer can just rerun the same process as needed and incorporate a sequence number in the values. The distinguished proposer is, of course, the single point of failure but an election mechanism can select new proposers.

(revised)

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