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 submitsvote(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 votevote(X', S')
where the range[X, S.last]
overlaps the range[X', S'.last]
,X != X'
, as shown below:
++
 
++ ++
   
 ++ 
 
 
 
+++ 
  
X   
  
+++ 
 
 +++
  
   X'
  
 +++
 
 
 
 
 +++
  
   S'.last
  
 ++

+++
 
s  
 
+++




+++
 
S.last  
 
++
(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
inS
are ancestors/descendants of one another, alls'
inS'
are ancsestors/descendants of one another, X == X'
impliesS
is parent ofS'
orS'
is a parent ofS
X' > X
impliesX' > S.last
andS'.last > S.last
and for alls
inS
,s + lockout(s) < X'
X > X'
impliesX > S'.last
andS.last > S'.last
and for alls
inS'
,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:
The sum of the stakes of all the validator id's
> 1/3
For each
(validator_id, validator_vote(X, S))
, there exists some slots
inS
where: a.s
is not a common ancestor of bothvalidator_vote.last
andold_vote.last
andnew_vote.last
. b.s
is not a descendant ofvalidator_vote.last
. * c.s + s.lockout() >= old_vote.last
(implies validator is still locked out on slots
at slotold_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 ofB
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 ofB
,
then X > B
.
++
 
++ ++
   
 ++ 
 
 
 
 +++
  
   X'
  
 +++
 
 
 +++
  
   B (Optimistically Confirmed)
  
 +++
 
 
 
 +++
  
   S'.last
  
 ++

+++
 
X  
 
+++






+++
 
S.last  
 
+++




+++
 
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
++
 
++ ++
   
 ++ 
 
 
 
 +++
  
   X
  
 +++
 
 
 +++
  
   B (Optimistically Confirmed)
  
 +++
 
 
 
 +++
  
   S.last
  
 ++

+++
 
B'(Finalized)  
 
++
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 ofX
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 ofB'
 Also submitted a vote
v
of the formVote(X, S)
whereX <= 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 ofB'
, so it can't be a descendant ofB
 Because
S_v.last
is not a descendant ofB
, thenX_v
cannot be a descendant or ancestor ofB
.
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 ofS.last
andX'
s
is not a descendant ofS.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 ofB
andX'
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
.