In this figure, the proposal of X is rejected.
At the end of the timeline, S1 and S2 accept X while S3, S4 and S5 accept Y. Proposer X is now supposed to re-send the proposal with value Y.
But what happens if proposer X gets down at that time? How does S1 and S2 eventually learn the value Y?
Thanks in advance!
It is a little hard to answer this from the fragment of a diagram that you've shared since it is not clear what exactly it means. It would be helpful if you could link to the source of that diagram so we can see more of the context of your question. The rest of this answer is based on a guess as to its meaning.
There are three distinct roles in Paxos, commonly known as proposer, acceptor and learner, and I think it aids understanding to divide things into these three roles. The diagram you've shared looks like it is illustrating a set of five acceptors and the messages that they have sent as part of the basic Synod algorithm (a.k.a. single-instance Paxos). In general there's no relationship between the sets of learners and acceptors in a system: there might be a single learner, or there might be thousands, and I think it helps to separate these concepts out. Since S1 and S2 are acceptors, not learners, it doesn't make sense to ask about them learning a value. It is, however, valid to ask about how to deal with a learner that didn't learn a value.
In practical systems there is usually also another role of leader which takes responsibility for pushing the system forward using timeouts and retries and fault detectors and so on, to ensure that all learners eventually learn the chosen value or die trying, but this is outside the scope of the basic algorithm that seems to be illustrated here. In other words, this algorithm guarantees safety ("nothing bad happens") but does not guarantee liveness ("something good happens"). It is acceptable here if some of the learners never learn the chosen value.
The leader can do various things to ensure that all learners eventually learn the chosen value. One of the simplest strategies is to get the learned value from any learner and broadcast it to the other learners, which is efficient and works as long as there is at least one running learner that's successfully learned the chosen value. If there is no such learner, the leader can trigger another round of the algorithm, which will normally result in the chosen value being learned. If it doesn't then its only option is to retry, and keep retrying until eventually one of these rounds succeeds.
In this figure, the proposal of X is rejected.
My reading of the diagram is that it is an ”accept request” that is rejected. Page 5 paragraph 1 of Paxos Made Simple describes this message type.
Proposer X is now supposed to re-send the proposal with value Y.
The diagram does not indicate that. Only if Y was seen in response to the blue initial propose messages would the blue proposer have to choose Y. Yet the blue proposer chose X as the value in its ”accept request”. If it is properly following Paxos it could not have ”seen Y” in response to it's initial proposal message. If it had seen it then it must have chosen it and so it wouldn’t have sent X.
In order to really know what is happening you would need to know what responses were seen by each proposer. We cannot see from the diagram what values, if any, were returned in response to the first three blue propose messages. We don’t see in the diagram whether X was previously accepted at any node or whether it was not. We don't know if the blue proposer was ”free to choose” it's own X or had to use an X that was already accepted at one or more nodes.
But what happens if proposer X gets down at that time?
If the blue proposer dies then this is not a problem. The green proposer has successfully fixed the value Y at a majority of the nodes.
How does S1 and S2 eventually learn the value Y?
The more interesting scenario is what happens if the green proposer dies. The green proposer may have sent it's accept request messages containing Y and immediately died. As three of the messages are successful the value Y has been fixed yet the original proposer may not be alive to see the accept response messages. For any further progress to be made a new proposer needs to send a new propose message. As three of the nodes will reply with Y the new proposer will chose Y as the value of it's accept request message. This will be sent to all nodes and if all messages get through, and no other proposer interrupts, then S1 and S2 will become consistent.
The essence of the algorithm is collaboration. If a proposer dies the next proposer will collaborate and chose the highest value previously proposed if any exists.
