Optimistic Confirmation

Primitives

vote(X, S) - Votes will be augmented with a "reference" slot, X which is the latest ancestor of this fork that this validator voted on with a proof of switching. As long as the validator makes consecutive votes that are all descended from each other, the same X should be used for all those votes. When the validator makes a vote for a slot s that is not descended from the previous, X will be set to the new slot s. All votes will then be of the form vote(X, S), where S is the sorted list of slots (s, s.lockout) being voted for.

Given a vote vote(X, S), let S.last == vote.last be the last slot in S.

Now we define some "Optimistic Slashing" slashing conditions. The intuition for these is described below:

  • Intuition: If a validator submits vote(X, S), the same validator should not have voted on a different fork that "overlaps" this fork. More concretely, this validator should not have cast another vote vote(X', S') where the range [X, S.last] overlaps the range [X', S'.last], X != X', as shown below:
+-------+
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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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| +---+---+
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| | | S'.last
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| +-------+
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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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+-------+

(Example of slashable votes vote(X', S') and vote(X, S))

In the diagram above, note that the vote for S.last must have been sent after the vote for S'.last (due to lockouts, the higher vote must have been sent later). Thus, the sequence of votes must have been: X ... S'.last ... S.last. This means after the vote on S'.last, the validator must have switched back to the other fork at some slot s > S'.last > X. Thus, the vote for S.last should have used s as the "reference" point, not X, because that was the last "switch" on the fork.

To enforce this, we define the "Optimistic Slashing" slashing conditions. Given any two distinct votes vote(X, S)and vote(X', S') by the same validator, the votes must satisfy:

  • X <= S.last, X' <= S'.last
  • All s in S are ancestors/descendants of one another, all s' in S' are ancsestors/descendants of one another,
  • X == X' implies S is parent of S' or S' is a parent of S
  • X' > X implies X' > S.last and S'.last > S.last and for all s in S, s + lockout(s) < X'
  • X > X' implies X > S'.last and S.last > S'.last and for all s in S', s + lockout(s) < X

(The last two rules imply the ranges cannot overlap): Otherwise the validator is slashed.

Range(vote) - Given a vote v = vote(X, S), define Range(v) to be the range of slots [X, S.last].

SP(old_vote, new_vote) - This is the "Switching Proof" for old_vote, the validator's latest vote. Such a proof is necessary anytime a validator switches their "reference" slot (see vote section above). The switching proof includes a reference to old_vote, so that there's a record of what the "range" of that old_vote was (to make other conflicting switches in this range slashable). Such a switch must still respect lockouts.

A switching proof shows that > 1/3 of the network is locked out at slot old_vote.last.

The proof is a list of elements (validator_id, validator_vote(X, S)), where:

  1. The sum of the stakes of all the validator id's > 1/3

  2. For each (validator_id, validator_vote(X, S)), there exists some slot s in S where: a.s is not a common ancestor of both validator_vote.last and old_vote.last and new_vote.last. b. s is not a descendant of validator_vote.last. * c. s + s.lockout() >= old_vote.last (implies validator is still locked out on slot s at slot old_vote.last).

Switching forks without a valid switching proof is slashable.

Definitions:

Optimistic Confirmation - A block B is then said to have achieved "optimistic confirmation" if >2/3 of stake have voted with votes v where Range(v) for each such v includes B.slot.

Finalized - A block B is said to be finalized if at least one correct validator has rooted B or a descendant of B.

Reverted - A block B is said to be reverted if another block B' that is not a parent or descendant of B was finalized.

Guarantees:

A block B that has reached optimistic confirmation will not be reverted unless at least one validator is slashed.

Proof:

Assume for the sake of contradiction, a block B has achieved optimistic confirmation at some slot B + n for some n, and:

  • Another block B' that is not a parent or descendant of B was finalized.
  • No validators violated any slashing conditions.

By the definition of optimistic confirmation, this means > 2/3 of validators have each shown some vote v of the form Vote(X, S) where X <= B <= v.last. Call this set of validators the Optimistic Validators.

Now given a validator v in Optimistic Validators, given two votes made by v, Vote(X, S) and Vote(X', S') where X <= B <= S.last, and X' <= B <= S'.last, then X == X' otherwise an "Optimistic Slashing" condition is violated (the "ranges" of each vote would overlap at B).

Thus define the Optimistic Votes to be the set of votes made by Optimistic Validators, where for each optimistic validator v, the vote made by v included in the set is the maximal vote Vote(X, S) with the greatest S.last out of any votes made by v that satisfy X <= B <= S.last. Because we know from above X for all such votes made by v is unique, we know there is such a unique maximal vote.

Lemma 1:

Claim: Given a vote Vote(X, S) made by a validator V in the Optimistic Validators set, and S contains a vote for a slot s for which:

  • s + s.lockout > B,
  • s is not an ancestor or descendant of B,

then X > B.

+-------+
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+---------+ +--------+
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| | | B (Optimistically Confirmed)
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| +---+---+
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| | | S'.last
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| +-------+
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+---+---+
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X | |
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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 | |
+-------+

Proof: Assume for the sake of contradiction a validator V from the "Optimistic Validators" set made such a vote Vote(X, S) where S contains a vote for a slot s not an ancestor or descendant of B, where s + s.lockout > B, but X <= B.

Let Vote(X', S') be the vote in Optimistic Votes set made by validator V. By definition of that set (all votes optimistically confirmed B), X' <= B <= S'.last (see diagram above).

This implies that because it's assumed above X <= B, then X <= S'.last, so by the slashing rules, either X == X' or X < X' (otherwise would overlap the range (X', S'.last)).

Case X == X':

Consider s. We know s != X because it is assumed s is not an ancestor or descendant of B, and X is an ancestor of B. Because S'.last is a descendant of B, this means s is also not an ancestor or descendant of S'.last. Then because S.last is descended from s, then S'.last cannot be an ancestor or descendant of S.last either. This implies X != X' by the "Optimistic Slashing" rules.

Case X < X':

Intuitively, this implies that Vote(X, S) was made "before" Vote(X', S').

From the assumption above, s + s.lockout > B > X'. Because s is not an ancestor of X', lockouts would have been violated when this validator first attempted to submit a switching vote to X' with some vote of the form Vote(X', S'').

Since none of these cases are valid, the assumption must have been invalid, and the claim is proven.

Lemma 2:

Recall B' was the block finalized on a different fork than "optimistically" confirmed" block B.

Claim: For any vote Vote(X, S) in the Optimistic Votes set, it must be true that B' > X

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| | | X
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| | | B (Optimistically Confirmed)
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| +---+---+
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| | | S.last
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| +-------+
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+---+---+
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B'(Finalized) | |
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+-------+

Proof: Let Vote(X, S) be a vote in the Optimistic Votes set. Then by definition, given the "optimistcally confirmed" block B, X <= B <= S.last.

Because X is a parent of B, and B' is not a parent or ancestor of B, then:

  • B' != X
  • B' is not a parent of X

Now consider if B' < X:

Case B' < X: We wll show this is a violation of lockouts. From above, we know B' is not a parent of X. Then because B' was rooted, and B' is not a parent of X, then the validator should not have been able to vote on the higher slot X that does not descend from B'.

Proof of Safety:

We now aim to show at least one of the validators in the Optimistic Validators set violated a slashing rule.

First note that in order for B' to have been rooted, there must have been > 2/3 stake that voted on B' or a descendant of B'. Given that the Optimistic Validator set also contains > 2/3 of the staked validators, it follows that > 1/3 of the staked validators:

  • Rooted B' or a descendant of B'
  • Also submitted a vote v of the form Vote(X, S) where X <= B <= v.last.

Let the Delinquent set be the set of validators that meet the above criteria.

By definition, in order to root B', each validator V in Delinquent must have each made some "switching vote" of the form Vote(X_v, S_v) where:

  • S_v.last > B'
  • S_v.last is a descendant of B', so it can't be a descendant of B
  • Because S_v.last is not a descendant of B, then X_v cannot be a descendant or ancestor of B.

By definition, this delinquent validator V also made some vote Vote(X, S) in the Optimistic Votes where by definition of that set (optimistically confirmed B), we know S.last >= B >= X.

By Lemma 2 we know B' > X, and from above S_v.last > B', so then S_v.last > X. Because X_v != X (cannot be a descendant or ancestor of B from above), then by the slashing rules then, we know X_v > S.last. From above, S.last >= B >= X so for all such "switching votes", X_v > B.

Now ordering all these "switching votes" in time, let V to be the validator in Optimistic Validators that first submitted such a "swtching vote" Vote(X', S'), where X' > B. We know that such a validator exists because we know from above that all delinquent validators must have submitted such a vote, and the delinquent validators are a subset of the Optimistic Validators.

Let Vote(X, S) be the unique vote in Optimistic Votes made by validator V (maximizing S.last).

Given Vote(X, S) because X' > B >= X, then X' > X, so by the "Optimistic Slashing" rules, X' > S.last.

In order to perform such a "switching vote" to X', a switching proof SP(Vote(X, S), Vote(X', S')) must show > 1/3 of stake being locked out at this validator's latest vote, S.last. Combine this >1/3 with the fact that the set of validators in the Optimistic Voters set consists of > 2/3 of the stake, implies at least one optimistic validator W from the Optimistic Voters set must have submitted a vote (recall the definition of a switching proof),Vote(X_w, S_w) that was included in validator V's switching proof for slot X', where S_w contains a slot s such that:

  • s is not a common ancestor of S.last and X'
  • s is not a descendant of S.last.
  • s' + s'.lockout > S.last

Because B is an ancestor of S.last, it is also true then:

  • s is not a common ancestor of B and X'
  • s' + s'.lockout > B

which was included in V's switching proof.

Now because W is also a member of Optimistic Voters, then by the Lemma 1 above, given a vote by W, Vote(X_w, S_w), where S_w contains a vote for a slot s where s + s.lockout > B, and s is not an ancestor of B, then X_w > B.

Because validator V included vote Vote(X_w, S_w) in its proof of switching for slot X', then his implies validator V' submitted vote Vote(X_w, S_w) before validator V submitted its switching vote for slot X', Vote(X', S').

But this is a contradiction because we chose Vote(X', S') to be the first vote made by any validator in the Optimistic Voters set where X' > B and X' is not a descendant of B.