409 lines
17 KiB
Markdown
409 lines
17 KiB
Markdown
---
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title: Optimistic Confirmation
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---
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## Primitives
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`vote(X, S)` - Votes will be augmented with a "reference" slot, `X`
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which is the **latest** ancestor of this fork that this validator voted on
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with a proof of switching. As long as the validator makes consecutive votes
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that are all descended from each other, the same `X` should be used for all
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those votes. When the validator makes a vote for a slot `s` that is not
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descended from the previous, `X` will be set to the new slot `s`. All votes
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will then be of the form `vote(X, S)`, where `S` is the sorted list of slots
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`(s, s.lockout)` being voted for.
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Given a vote `vote(X, S)`, let `S.last == vote.last` be the last slot in `S`.
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Now we define some "Optimistic Slashing" slashing conditions. The intuition
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for these is described below:
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- `Intuition`: If a validator submits `vote(X, S)`, the same validator
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should not have voted on a different fork that "overlaps" this fork.
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More concretely, this validator should not have cast another vote
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`vote(X', S')` where the range `[X, S.last]` overlaps the range
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`[X', S'.last]`, `X != X'`, as shown below:
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```text
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+-------+
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+---------+ +--------+
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+---+---+ |
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X | | |
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+---+---+ |
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| | | X'
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| +---+---+
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| | | S'.last
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+---+---+
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s | |
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+---+---+
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+---+---+
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S.last | |
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+-------+
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```
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(Example of slashable votes vote(X', S') and vote(X, S))
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In the diagram above, note that the vote for `S.last` must have been sent after
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the vote for `S'.last` (due to lockouts, the higher vote must have been sent
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later). Thus, the sequence of votes must have been: `X ... S'.last ... S.last`.
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This means after the vote on `S'.last`, the validator must have switched back
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to the other fork at some slot `s > S'.last > X`. Thus, the vote for `S.last`
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should have used `s` as the "reference" point, not `X`, because that was the
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last "switch" on the fork.
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To enforce this, we define the "Optimistic Slashing" slashing conditions. Given
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any two distinct votes `vote(X, S)`and `vote(X', S')` by the same validator,
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the votes must satisfy:
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- `X <= S.last`, `X' <= S'.last`
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- All `s` in `S` are ancestors/descendants of one another,
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all `s'` in `S'` are ancsestors/descendants of one another,
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-
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- `X == X'` implies `S` is parent of `S'` or `S'` is a parent of `S`
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- `X' > X` implies `X' > S.last` and `S'.last > S.last`
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and for all `s` in `S`, `s + lockout(s) < X'`
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- `X > X'` implies `X > S'.last` and `S.last > S'.last`
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and for all `s` in `S'`, `s + lockout(s) < X`
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(The last two rules imply the ranges cannot overlap):
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Otherwise the validator is slashed.
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`Range(vote)` - Given a vote `v = vote(X, S)`, define `Range(v)` to be the range
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of slots `[X, S.last]`.
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`SP(old_vote, new_vote)` - This is the "Switching Proof" for `old_vote`, the
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validator's latest vote. Such a proof is necessary anytime a validator switches
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their "reference" slot (see vote section above). The switching proof includes
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a reference to `old_vote`, so that there's a record of what the "range" of that
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`old_vote` was (to make other conflicting switches in this range slashable).
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Such a switch must still respect lockouts.
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A switching proof shows that `> 1/3` of the network is locked out at slot
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`old_vote.last`.
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The proof is a list of elements `(validator_id, validator_vote(X, S))`, where:
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1. The sum of the stakes of all the validator id's `> 1/3`
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2. For each `(validator_id, validator_vote(X, S))`, there exists some slot `s`
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in `S` where:
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_ a.`s` is not a common ancestor of both `validator_vote.last` and
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`old_vote.last` and `new_vote.last`.
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_ b. `s` is not a descendant of `validator_vote.last`. \* c. `s + s.lockout() >= old_vote.last` (implies validator is still locked
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out on slot `s` at slot `old_vote.last`).
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Switching forks without a valid switching proof is slashable.
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## Definitions:
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Optimistic Confirmation - A block `B` is then said to have achieved
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"optimistic confirmation" if `>2/3` of stake have voted with votes `v`
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where `Range(v)` for each such `v` includes `B.slot`.
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Finalized - A block `B` is said to be finalized if at least one
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correct validator has rooted `B` or a descendant of `B`.
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Reverted - A block `B` is said to be reverted if another block `B'` that
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is not a parent or descendant of `B` was finalized.
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## Guarantees:
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A block `B` that has reached optimistic confirmation will not be reverted
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unless at least one validator is slashed.
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## Proof:
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Assume for the sake of contradiction, a block `B` has achieved
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`optimistic confirmation` at some slot `B + n` for some `n`, and:
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- Another block `B'` that is not a parent or descendant of `B`
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was finalized.
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- No validators violated any slashing conditions.
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By the definition of `optimistic confirmation`, this means `> 2/3` of validators
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have each shown some vote `v` of the form `Vote(X, S)` where `X <= B <= v.last`.
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Call this set of validators the `Optimistic Validators`.
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Now given a validator `v` in `Optimistic Validators`, given two votes made by
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`v`, `Vote(X, S)` and `Vote(X', S')` where `X <= B <= S.last`, and
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`X' <= B <= S'.last`, then `X == X'` otherwise an "Optimistic Slashing" condition
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is violated (the "ranges" of each vote would overlap at `B`).
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Thus define the `Optimistic Votes` to be the set of votes made by
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`Optimistic Validators`, where for each optimistic validator `v`, the vote made
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by `v` included in the set is the `maximal` vote `Vote(X, S)` with the
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greatest `S.last` out of any votes made by `v` that satisfy `X <= B <= S.last`.
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Because we know from above `X` for all such votes made by `v` is unique, we know
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there is such a unique `maximal` vote.
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### Lemma 1:
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`Claim:` Given a vote `Vote(X, S)` made by a validator `V` in the
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`Optimistic Validators` set, and `S` contains a vote for a slot `s`
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for which:
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- `s + s.lockout > B`,
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- `s` is not an ancestor or descendant of `B`,
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then `X > B`.
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```text
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+-------+
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+---------+ +--------+
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| +---+---+
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| | | X'
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| | | B (Optimistically Confirmed)
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| | | S'.last
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+---+---+
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+---+---+
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+---+---+
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S.last | |
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+---+---+
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+---+---+
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s + s.lockout | |
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+-------+
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```
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`Proof`: Assume for the sake of contradiction a validator `V` from the
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"Optimistic Validators" set made such a vote `Vote(X, S)` where `S` contains
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a vote for a slot `s` not an ancestor or descendant of `B`, where
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`s + s.lockout > B`, but `X <= B`.
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Let `Vote(X', S')` be the vote in `Optimistic Votes` set made by validator `V`.
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By definition of that set (all votes optimistically confirmed `B`),
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`X' <= B <= S'.last` (see diagram above).
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This implies that because it's assumed above `X <= B`, then `X <= S'.last`,
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so by the slashing rules, either `X == X'` or `X < X'` (otherwise would
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overlap the range `(X', S'.last)`).
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`Case X == X'`:
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Consider `s`. We know `s != X` because it is assumed `s` is not an ancestor
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or descendant of `B`, and `X` is an ancestor of `B`. Because `S'.last` is a
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descendant of `B`, this means `s` is also not an ancestor or descendant of
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`S'.last`. Then because `S.last` is descended from `s`, then `S'.last` cannot
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be an ancestor or descendant of `S.last` either. This implies `X != X'` by the
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"Optimistic Slashing" rules.
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`Case X < X'`:
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Intuitively, this implies that `Vote(X, S)` was made "before" `Vote(X', S')`.
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From the assumption above, `s + s.lockout > B > X'`. Because `s` is not an
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ancestor of `X'`, lockouts would have been violated when this validator
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first attempted to submit a switching vote to `X'` with some vote of the
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form `Vote(X', S'')`.
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Since none of these cases are valid, the assumption must have been invalid,
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and the claim is proven.
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### Lemma 2:
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Recall `B'` was the block finalized on a different fork than
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"optimistically" confirmed" block `B`.
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`Claim`: For any vote `Vote(X, S)` in the `Optimistic Votes` set, it must be
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true that `B' > X`
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```text
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+-------+
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+--------+ +---------+
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| +---+---+
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| +---+---+
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| +---+---+
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| | | B (Optimistically Confirmed)
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| +---+---+
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| +-------+
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+---+---+
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B'(Finalized) | |
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+-------+
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```
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`Proof`: Let `Vote(X, S)` be a vote in the `Optimistic Votes` set. Then by
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definition, given the "optimistcally confirmed" block `B`, `X <= B <= S.last`.
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Because `X` is a parent of `B`, and `B'` is not a parent or ancestor of `B`,
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then:
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- `B' != X`
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- `B'` is not a parent of `X`
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Now consider if `B'` < `X`:
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`Case B' < X`: We wll show this is a violation of lockouts.
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From above, we know `B'` is not a parent of `X`. Then because `B'` was rooted,
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and `B'` is not a parent of `X`, then the validator should not have been able
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to vote on the higher slot `X` that does not descend from `B'`.
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### Proof of Safety:
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We now aim to show at least one of the validators in the
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`Optimistic Validators` set violated a slashing rule.
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First note that in order for `B'` to have been rooted, there must have been
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`> 2/3` stake that voted on `B'` or a descendant of `B'`. Given that the
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`Optimistic Validator` set also contains `> 2/3` of the staked validators,
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it follows that `> 1/3` of the staked validators:
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- Rooted `B'` or a descendant of `B'`
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- Also submitted a vote `v` of the form `Vote(X, S)` where `X <= B <= v.last`.
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Let the `Delinquent` set be the set of validators that meet the above
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criteria.
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By definition, in order to root `B'`, each validator `V` in `Delinquent`
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must have each made some "switching vote" of the form `Vote(X_v, S_v)` where:
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- `S_v.last > B'`
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- `S_v.last` is a descendant of `B'`, so it can't be a descendant of `B`
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- Because `S_v.last` is not a descendant of `B`, then `X_v` cannot be a
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descendant or ancestor of `B`.
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By definition, this delinquent validator `V` also made some vote `Vote(X, S)`
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in the `Optimistic Votes` where by definition of that set (optimistically
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confirmed `B`), we know `S.last >= B >= X`.
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By `Lemma 2` we know `B' > X`, and from above `S_v.last > B'`, so then
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`S_v.last > X`. Because `X_v != X` (cannot be a descendant or ancestor of
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`B` from above), then by the slashing rules then, we know `X_v > S.last`.
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From above, `S.last >= B >= X` so for all such "switching votes", `X_v > B`.
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Now ordering all these "switching votes" in time, let `V` to be the validator
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in `Optimistic Validators` that first submitted such a "swtching vote"
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`Vote(X', S')`, where `X' > B`. We know that such a validator exists because
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we know from above that all delinquent validators must have submitted such
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a vote, and the delinquent validators are a subset of the
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`Optimistic Validators`.
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Let `Vote(X, S)` be the unique vote in `Optimistic Votes` made by
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validator `V` (maximizing `S.last`).
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Given `Vote(X, S)` because `X' > B >= X`, then `X' > X`, so
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by the "Optimistic Slashing" rules, `X' > S.last`.
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In order to perform such a "switching vote" to `X'`, a switching proof
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`SP(Vote(X, S), Vote(X', S'))` must show `> 1/3` of stake being locked
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out at this validator's latest vote, `S.last`. Combine this `>1/3` with the
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fact that the set of validators in the `Optimistic Voters` set consists of
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`> 2/3` of the stake, implies at least one optimistic validator `W` from the
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`Optimistic Voters` set must have submitted a vote (recall the definition of
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a switching proof),`Vote(X_w, S_w)` that was included in validator `V`'s
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switching proof for slot `X'`, where `S_w` contains a slot `s` such that:
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- `s` is not a common ancestor of `S.last` and `X'`
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- `s` is not a descendant of `S.last`.
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- `s' + s'.lockout > S.last`
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Because `B` is an ancestor of `S.last`, it is also true then:
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- `s` is not a common ancestor of `B` and `X'`
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- `s' + s'.lockout > B`
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which was included in `V`'s switching proof.
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Now because `W` is also a member of `Optimistic Voters`, then by the `Lemma 1`
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above, given a vote by `W`, `Vote(X_w, S_w)`, where `S_w` contains a vote for
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a slot `s` where `s + s.lockout > B`, and `s` is not an ancestor of `B`, then
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`X_w > B`.
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Because validator `V` included vote `Vote(X_w, S_w)` in its proof of switching
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for slot `X'`, then his implies validator `V'` submitted vote `Vote(X_w, S_w)`
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**before** validator `V` submitted its switching vote for slot `X'`,
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`Vote(X', S')`.
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But this is a contradiction because we chose `Vote(X', S')` to be the first vote
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made by any validator in the `Optimistic Voters` set where `X' > B` and `X'` is
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not a descendant of `B`.
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