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\title{Polkadot Finality Gadget v9000}
\date{\today}
\maketitle

We have a chain selection rule and block production mechanism that probably gives us consensus on a prefix of the chain including all but the most recent blocks. However we need to prove to 3rd parties that far enough back blocks are final by producing a certificate signed by $2/3$ of validators.

The idea is to run a BFT agreement algorithm similar to Tendermint or Algorand agreement but on chains rather than individual blocks so we do not need to reach agreement on each block. The algorithm will have a primary but their role will be to coordinate locking rather than propose blocks. Instead everyone will vote on their best block.

We can consider a prevote or precommit for a block as a vote for every block on that chain not already finalised. Thus we may have $2/3$ votes on many blocks in one vote but it is easy to see that all blocks with $2/3$ votes are in one chain.

We will want to change the set of participants who actively agree sometimes. To model this, we have a large set of participants who follow messages. For each voting step, there is a set of $n$ voters. We will frequently need to assume that for each such step, at most  $f < n/3$ voters are faulty. We need $n-f$ of voters to agree on finality. Whether or not block producers ever vote, they will need to be participants who track the state of the protocol.

Participants remember which block they see as currently being the latest finalised block and a chain they are locked to. This locked chain represents an estimate of which block could have been finalised already.

{\bf Rounds}: each participant has their own idea of what the current round number is. Every prevote and precommit has an associated round number. Honest voters only vote once (for each type of vote) in each round and don't vote in earlier rounds after later ones.

Each round has two phases, each of which has an associated vote, prevote and precommit.

\section{ Preliminaries }

For block $B$, we write $\mathrm{chain}(B)$ for the chain whose head is $B$. The block number, $n(B)$ of a block $B$ is the length of $\mathrm{chain}(B)$.

For blocks $B'$, $B$, $B$ is later than $B'$ if it has a higher block number.
We write $B > B'$ or that $B$ is descendant of $B'$ for $B$, $B'$ appearing in the same blockchain with $B'$ later i.e. $B \in \mathrm{chain}(B')$ with $n(B') > n(B)$ and $B < B'$ or $B$ is an ancestor of $B'$ for $B' \in \mathrm{chain}(B)$ with $n(B) > n(B')$ . $B \geq B'$ an $B \leq B'$ are similar except allowing $B = B$. We write $B \sim B'$ or $B$ and $B'$ are on the same chain if $B<B'$,$B=B'$ or $B> B'$ and $B \nsim B'$ or $B$ and $B'$ are not on the same chain if there is no such chain.

Blocks are ordered as a tree with the genesis block as root. So any two blocks have a common ancestor but two blocks not on the same chain do not have a common descendant.

A vote $v$ for a block $B$ by a validator $V$ is a message signed by $V$ containing the blockhash of $B$ and meta information like the round numbers and the type of vote. 

We call a set $S$ of votes tolerant if the number of validators with more than one vote in $S$ is at most $f$. We say that $S$ has supermajority for a block $B$ if the set of validators with votes for blocks $\geq B$ has size at least $(n+f+1)/2$.

The $2/3$-GHOST function $g(S)$ takes a set $S$ of votes and returns the block $B$ with highest block number such that $S$ has a supermajority for $B$. If there is no such block, then it returns `nil`. (if $f \neq \lfloor (n-1)/3 \rfloor$, then this is a misnomer and we may change the name accordingly.)

Note that, if $S$ is tolerant, then we can compute $g(S)$ by starting at the genesis block and iteratively looking for a child of our current block with a supermajority, which must be unique if it exists. Thus we have:
\begin{lemma} \label{lem:ghost-monotonicity}
Let $T$ be a tolerant set of votes. Then
\begin{enumerate}
\item The above definition uniquely defines $g(T)$
\item If $S \subseteq T$ has $g(S) \neq$ nil, then $g(S) \leq g(T)$.
\item If $S_i \subseteq T$ for $1 \leq i \leq n$ then all non-nil $g(S_i)$ are on a single chain with head $g(T)$.
\end{enumerate}

\end{lemma}

Note that we can easily update $g(S)$ to $g(S \cup \{v\})$, by checking if any child of $g(S)$ now has a supermajority.

3 tells us that even if validators see different subsets of the votes cast in a given voting round, this rule may give them different blocks but all such blocks are in the same chain under this assumption. 

We say that it is possible for a set $S$ to have a supermajority for $B$ if $2f+1$ validators either vote for a block $\not \geq B$ or equivocate in $S$. Note that if $S$ is tolerant, it is possible for $S$ to have a supermajority for $B$ if and only if there is a tolerant $T \supseteq S$ that has a supermajority for $B$.


We say that it is impossible for any child of $B$ to have a supermajority in $S$ if $S$ has votes from at least $2f+1$ validators and it is impossible for $S$ to have a supermajority for each child of $B$ appearing on the chain of any vote in $S$. Again, provided $S$ is tolerant, this holds if and only if for any possible child of $B$, there is no tolerant $T \subset S$ that has a supermajority for that child.

Note that it is possible for an intolerant $S$ to both have a supermajority for $S$ and for it to be impossible to have such a supermajority under these definitions, as we regard such sets as impossible anyway.

\begin{lemma} \label{lem:impossible}
\begin{itemize}
\item[(i)] If $B' \geq B$ and it is impossible for $S$ to have a supermajority for $B$, then it is impossible for $S$ to have a supermajority for $B'$.
\item[(ii)] If $S \subseteq T$ and it is impossible for $S$ to have a supermajority for $B$
\item[(iii)] If $g(S)$ exists and $B \nsim g(S)$ then it is impossible for $S$ to have a supermajority for $B$.
\end{itemize}
\end{lemma}

\section{Algorithm}

We let $V_{r,v}$ and $C_{r,v}$ be the sets of prevotes and precommits respectively received by $v$ from round $r$ at the current time.

We define $E_{r,v}$, $v$'s estimate of what might have been finalised in round $r$, to be the last block in the chain with head $g(V_{r,v})$ that it is possible for $C_{r,r}$ to have a supermajority for. If either $E_{r,v} < g(V_{r,v})$ or it is impossible for $C_{r,v}$ to have a supermajority for any children of $g(V_{r,v})$,, then we say that ($v$ sees that) round $r$ is completable. $E_{0,v}$ is the genesis block (if we start at $r=1$).

We have a time bound $T$, that we hope is enough to send messages and gossip them to everyone. 

In round $r$ an honest validator $v$ does the following:

1. $v$ can start round $r > 1$ when round $r-1$ is completable and $v$ has cast votes in all previous rounds where they are a voter. Let $t_{r,v}$ be the time we start round $r$.

2. At time $t_{r,v}$, if $v$ is the primary of this round and has not finalised $E_{r-1,v}$ then they broadcast $E_{r-1,v}$. If thy have finalised it, they can broadcast $E_{r-1,v}$ anyway (but don't need to).

3. If $v$ is a voter for the prevote of round $r$, $v$ waits until either it is at least time $t_{r,v}+2T$ or round $r$ is completable, then broadcasts a prevote. They prevote for the head of the best chain containing $E_{r-1,v}$ unless we received a block $B$ from the primary and $g(V_{r-1,v}) \geq B > E_{r-1,v}$, in which case they use the best chain containing $B$ instead.

4. If $v$ is a voter for the precommit step in round $r$, then they wait until $g(V_{r,v}) \geq E_{r-1,v}$ and one of the following  conditions holds
(i) it is at least time $t_{r,v}+4T$, 
(ii) round $r$ is completable or
(iii) it is impossible for $V_{r,v}$ to have a supermajority for any child of $g(V_{r,v})$,
and then broadcasts a precommit for $g(V_{r,v})$ {\em( (iii) is optional, we can get away with just (i) and (ii))}.


\subsection{ Finalisation}

If, for some round $r$, at any point after the precommit step of round $r$, we have that $B=g(C_{r,v})$ is later than our last finalised block and $V_{r,v}$ has a supermajority, then we finalise $B$. We may also send a commit message for $B$ that consists of $B$ and a set of precommits for blocks $\geq B$ (ideally for $B$ itself if possible see "Alternatives to the last blockhash" below). 

To avoid spam, we only send commit messages for $B$ if we have not receive any valid commit messages for $B$ and its descendants and we wait some time chosen uniformly at random from $[0,1]$ seconds or so before broadcasting.

If we receive a valid commit message for $B$ for round $r$, then it contains enough precommits to finalise $B$ itself if we haven't already done so, so we'll finalise $B$ as long as we are past the precommit step of round $r$.

\section{ Analysis }

\subsection{ Accountable Safety}

The first thing we want to show is asynchronous safety if we have at most $f$ Byzantine validators:

\begin{theorem} If the protocol finalises any two blocks $B,B'$ that have valid commit messages sent are on the same chain, then there are at least $f+1$ Byzantine voters who all voted in a particular vote. Furthermore, there is a synchronous procedure to find such a set.
\end{theorem}

The challenge procedure works as follows: If $B$ and $B'$ are committed in the same round, then the union of their precommits must contain at least $f$ equivocations, so we are done. Otherwise $B$ was committed in round $r$ and $B'$ in round $r' > r$ say. Then we ask the at least $n-f$ validators who precomitted $\geq B'$ in round $r$ in the commit message, why they precomitted.

We ask queries of the following form: 

- Why was $E_{r''-1} \not\geq B$  when you prevoted for or precomitted to $B'' \not\geq B$ in round $r'' > r$?

Which any honest validator should be able to respond to as is shown in Lemma \ref{lem:honest-answer} below. 

The response is of the following form:

- A either a set $S$ of prevotes for round $r''-1$ or a set $S$ of precommits for round $r''-1$  or  such that it is impossible for $S$ to have a supermajority for $B$.

If no validator responds, then we  have $n-f$ Byzantine validators. If any do, then if $r'' > r+1$,  we can ask the same query for $n-f$ validators in round $r''-1$.

If any responded and $r''=r+1$, then we have either a set $S$ of prevotes or precommits in round $r$ that it is impossible for $S$ to have a supermajority for $B$ in round $r$.

If $S$ is a set of precommits, then if we take the union of $S$ and the set of precommits in the commit message for $B$, then the resulting set of precommits for round $r$ has a supermajority for $B$ and it is impossible for it to have a supermajority for $B$. This is possible if the set is not tolerant and so there must be at least $f+1$ voters who equivocate an so are Byzantine.

If we get a set $S$ of prevotes for round $r$ that does not have a supermajority for $B$, then we need to ask a query of the form

- Which prevotes for round $r$ have you seen?

to all the voters of precommit in the commit message for $B$  who voted for blocks $B'' \geq B$. There must be $n-f$ such validators and a valid response to this query is a set $T$ of prevotes for round $r$ with a supermajority for $B''$ and so a supermajority for $B$.

If any give a valid response, by a similar argument to the above, $S \cup T$ will have $f+1$ equivocations.

So we either discover $f+1$ equivocations in a vote or else $n-f > f+1$ voters fail to validly respond like a honest voter could do to a query.


\begin{lemma} \label{lem:honest-answer}
An honest validator can answer the first type of query.
\end{lemma}
We first need to show that for any prevote or precommit in round $r$ cast by an honest validator $v$ for a block $B''$, at the time of the vote we had $B'' \geq E_{r-1,v}$. Prevotes should be for the head of a chain containing either $E_{r-1,v}$ or $g(V_{r,v})$ Since $g(V_{r,v}) \geq E_{r-1},v$, in either case we have $B'' \geq E_{r-1,v}$. Precommits should be for $g(V_{r,v})$ but $v$ waits until $g(V_{r,v}) \geq E_{r-1,v}$ before precommiting so again this holds.
Thus if $B'' \not\geq B$, then we had $E_{r-1,v} \not\geq B$.
Next we need to show that if we had $E_{r-1,v} \not\geq B$ at the time of the vote then we can respond to the query validly. 
If $B$ wasn't on the same chain with $g(V_{r-1,v})$, then by Lemma \ref{lem:impossible} (iii), it was impossible for $V_{r-1,v}$ to have a supermajority for $B$. If it was on the same chain as $g(V_{r,-1v})$, then it was on the same chain as $E_{r-1,v}$ as well.
Since $E_{r-1,v} \not\geq B$, in this case we must have $B > E_{r-1,v}$. However, possibly using that round $r-1$ is completable, it was impossible for $C_{r-1,v}$ to have a supermajority for any child of $E_{r-1,v}$ on the same chain with $g(V_{v,r})$ and in particular for the child of $E_{r-1,v}$ on $\textrm{chain}(B)$. By Lemma \ref{lem:impossible} (i), this means $C_{r-1,v}$ did not have a supermajority for $B$.

Thus we have that, at the time of th vote, for one of $V_{r-1,v}$, $C_{r-1,v}$, it was impossible to have a supermajority for $B$. The current sets $V_{r-1,v}$ and $C_{r-1,v}$ are supersets of those at the time of the vote, and so by Lemma \ref{lem:impossible} (ii), it is still impossible. Thus $v$ can respond validly.


This is enough to show Theorem 1. Not that if $v$ sees a commit message for a block $B$ in round $r$ and has that $E_{r',v} \not\geq B$, for some completable round $r' \geq r$, then they should also be able to start a challenge procedure that successfully identifies at least $f+1$ Byzantine voters in some round. Thus we have that:

\begin{lemma} \label{lem:overestimate-final}
If there at most $f$ Byzantine voters in any vote, $B$ was finalised in round $r$ and an honest participant $v$ sees that round $r' \geq r$ is completable, then $E_{r',v} \geq B$.
\end{lemma}

\subsection{Liveness }

We show the protocol is deadlock free and also that it finalises new blocks quickly in a weakly synchronous model.

Let's define $V_{r,v,t}$ be the set $V_{r,v}$ at time $t$ and similarly for $C_{r,v,t}$ and the block $E_{r,v,t}$ .


\begin{lemma} \label{lem:message-monotonicity-completed-estimate}
 Let $v,v'$ be (possibly identical) honest participants, $t,t'$ be times and $r$ be a round. Then if $V_{r,v,t} \subseteq V_{r,v',r'}$ and $C_{r,v,t} \subseteq C_{r,v',r'}$, all these sets are tolerant and $v$ sees that $r$ is completable at time $t$, then $E_{r,v,t} \leq E_{r,v',t'}$ and $v'$ sees that $r$ is completable at time $t'$.
\end{lemma}

\begin{proof} Since $v$ sees that $r$ is completable at time $t$, $V_{r,v,t}$, $C_{r,v,t}$ each contain votes from $n-f$ voters and so the same holds for $V_{r,v',t'}$ and $C_{r,v',t'}$. 
By Lemma \ref{lem:ghost-monotonicity}, $g(V_{r,v',t'}) \geq g(V_{r,v,t})$.
 Using Lemma \ref{lem:impossible}, since it is impossible for $C_{r,v,t}$ to have a supermajority for any children of $g(V_{r,v,t}$, it is impossible for $C_{r,v',t'})$ as well and so $E_{r,v',t'} \leq g(V_{r,v,t})$. But now $E_{r,v,t}$,$E_{r,v',t'}$ are the last blocks on $\textrm{chain}(g(V_{r,v,t}))$ that it is possible for $C_{r,v,t},C_{r,v',t'}$ respectively to have a supermajority for. Thus by Lemma \ref{lem:impossible} (ii), $E_{r,v',t'} \leq E_{r,v,t}$.
\end{proof}
 
\subsubsection{Deadlock Freeness}

Now we can show deadlock freeness for the asynchronous gossip network model, when a message that is sent or received by any honest participant is eventually received by all honest participants.

\begin{proposition} Suppose that we are in the asynchronous gossip network model and that at most $f$ voters for any vote are Byzantine. Then the protocol is deadlock free.\end{proposition}

\begin{proof} We need to show that if all honest participants reach some vote, then all of them eventually reach the next.

If all honest voters reach a vote, then they will vote and all honest participants see their votes. We need to deal with the two conditions that might block the algorithm even then. To reach the prevote of round $r$, a participant may be held up at the condition that round $r-1$ must be completable. To reach the precommit, a voter may be held up by the condition that $g(V_{r,v}) \geq E_{r-1,v}$.

For the first case, the prevote, let $S$ be the set of all prevotes from round $r-1$ that any honest voter saw before they precommitted in round $r-1$.
By Lemma \ref{lem:ghost-monotonicity}, when voter $v'$ precommitted, they do it for block $g(V_{r-1,v'}) \leq g(S)
$. Let $T$ be the set of precommits in round $r$ cast by honest voters. Then or any block $B \not\leq g(S)$, $T$ does not contain any votes that are $\geq B$ and so it is impossible for $T$ to have a supermajority for $B$. In particular, it is impossible for $T$ to have a supermajority for any child of $g(S)$. 

Now consider a voter $v$. By our network assumption, there is a time $t$ by which they have seen the votes in $S$ and $T$. Consider any $t' \geq t$. At this point we have $g(V_{r,v,t;}) \geq g(S)$. It is impossible for $C_{r,v,t'}$ to have a supermajority for any child of $g(S)$ and so $E_{r-1,v,t'} \leq g(S)$, whether or not this inequality is strict, we satisfy one of the two conditions for $v$ to see that round $r-1$ is completable at time $t'$. Thus if all honest voters reach the precommit vote of round $r-1$, all honest voters reach the prevote of round $r$.

Now we consider the second case, reaching the precommit. 
Note that any honest prevoter in round $r$ votes for a block $B_v \geq E_{r-1,v,t_v}$ where $t_v$ is the time they vote. Now consider any honest voter for the precommit $v'$. By some time $t'$, they have received all the messages received by each honest voter $v$ at time $t_v$ and $v'$'s prevote. 
Then by Lemma \ref{lem:overestimate-final}, $B_v \geq E_{r-1,v,t_v} \geq E_{r-1,v',t'}$. Since $V_{r,v',t'}$ contains these $B_v$, $g(V_{r,v',t'}) \geq  E_{r-1,v',t'}$. Thus if all honest voters prevote in round $r$, eventually all honest voters precommit in round $r$.

An easy induction completes the proof of the proposition.
\end{proof}

\subsubsection{Weakly synchronous liveness}

Now we consider the weakly synchronous gossip network model. The idea that there is some global stabilisation time($\textrm{GST}$) such that any message received or sent by an honest participant at time $t$ is received by all honest participants at time $\max\{t,\textrm{GST}\}+T$.

Let $t_r$ be the first time any honest participant enters round $r$ i.e. the minimum over honest participants $v$ of $t_{r,v}$.

\begin{lemma} \label{lem:timings}
Assume the weakly synchronous gossip network model and that each vote has at most $f$ Byzantine voters. Then if $t_r \geq \textrm{GST}$, we have that
\begin{itemize}
\item[(i)] $t_r \leq t_{r,v} \leq t_r+T$ for any honest participant $v$,
\item[(ii)] no honest voter prevotes before time $t_r+2T$,
\item[(iii)] any honest voter $v$ precommits at the latest at time $t_{r,v}+4T$,
\item[(iv)] for any honest $v$, $t_{r+1,v} \leq t_r + 6T$.
\end{itemize}
\end{lemma}


\begin{proof} Let $v'$ be one of the first validators to enter round $r$ i.e. with $t_{r,v'}=t_r$. 
By our network assumption, all messages received by $v'$ before they ended are received by all honest participants before time $t_r+T$.
 In particular at time $t_r$, $v'$ sees that all previous rounds are completable and so by Lemma \ref{lem:overestimate-final}, so does every other honest validator by time $t_r+T$. Also since for $r' < r$, at some time $s_{r'} \leq t_r$ $g(V_{r',v',s_r'}) \geq E_{r',v',s_r'}$, again by Lemma 4, for all honest $v$, $g(V_{r',v,t_r+T}) \geq E_{r',v,t_r+T}$. Looking at the conditions for voting, this means that any honest validator does not need to wait before voting in any round $r' \leq r$. 
 Thus they cast any remaining votes and enter round $r$ by time $t_r + T$. This shows (i).

For (ii), note that the only reason why an honest voter would not wait until time $t_{r,v}+2T \geq t_r+ 2T$ is when $n-f$ voters have already prevoted. But since some of those $n-f$ votes are honest, this is impossible before $t_r+2T$

Now an honest voter $v''$ prevotes at time $t_{r,v''}+2T \leq t_r +3T$ and by our network assumptions all honest validators receive this vote by time $t_r+4T$. An honest voter for the precommit $v$ has also received all messages that $v''$ received before they prevoted by then. Thus the block they prevoted has $B_{v''} \geq E_{r-1,v''} \geq E_{r-1,v,t_r+4T}$, since this holds for every honest voter $v''$, $g(V_{r,v,t_r+4T}) \geq E_{r-1,v,t_r+4T}$. Thus they will precommit by time $t_{r,v}+4T$ which shows (iii).

By the network assumption an honest voter $v'$'s precommit will be received by all honest validators $v$ by time $t_{r,v'}+ 5T \leq t_r+6T$. Since $v$ will also have received all prevotes $v$ say when they precommitted by this time, their vote $B_{v'}$ will have $B_{v'}=g(V_{r,v'}) \leq g(V_{r,v,t_r+6T})$. Thus $C_{r, v, t_r+6T}$ contains precommits from $n-f$ voters $v'$ with $B_{v'} \leq g(V_{r,v,t_r+6T})$ and thus it is impossible for $C_{r,v,t_r+6T}$ to have a supermajority for any children of $g(V_{r,v, t_r+6T})$. Thus $v$ sees that round $r$ is completable at time $t_r+6T$. Since they have already prevoted and precommitted if they were a voter, they will move to round $r+1$ by at latest $t_t+6T$. This is (iv).
\end{proof}

\begin{lemma} \label{lem:honest-prevote-timings}
Suppose $t_r \geq GST$ and very vote has at most $f$ Byzantine voters. Let $H_r$ be the set of prevotes ever cast by honest voters in round $r$. Then
\begin{itemize}
\item[(a)] any honest voter precommits to a block $\geq  g(H_r)$,

\item[(b)] every honest participant finalises $g(H_r)$ by time $t_r+6T$.
\end{itemize}
\end{lemma}

\begin{proof} For (a), we separate into cases based on which of the conditions (i)-(iii) that we wait for to precommit hold.

For (i), all honest voters prevote in round $r$ by time $t_r+3T$. So any honest voter $v$ who precommits at or after time $t_{r,v}+4T \geq t_r+4T$ has received all votes in $H_r$ and by Lemma \ref{lem:ghost-monotonicity}, precommits to a block $\geq g(H_r)$.

For (ii), we argue that no honest voter commits a block $\not\geq g(H_r)$ first. The result will then follow by an easy induction once the other cases are dealt with. Suppose that no honest voter has precommitted a block $\not \geq g(H_r)$ so far and that a voter $v$ votes early because of (ii).

Note that, since we assume that all precommits by honest voters so far were $\geq g(H_r)$, it is possible for $C_{r,v}$ to have a supermajority for $g(H_r)$. For (ii) to hold for a voter $v$ i.e for round $r$ to be completable, it must be the case that either it is impossible for $C_{r,v}$ to have a supermajority for $g(V_{r,v})$ or else be impossible for $C_{r,v}$ to have a supermajority for any children of $g(V_{r,v})$. By Lemma \ref{lem:impossible} cannot have $g(V_{r,v}) < g(H_r)$. But by Lemma \ref{lem:ghost-monotonicity}, these are on the same chain and so $g(V_{r,v}) \geq g(H_r)$. Since this is the block $v$ precommits to, we are done in case (ii)

For (iii), let $v$ be the voter in question. Note that since $n-f$ honest voters prevoted $\geq g(H_r)$, it is possible for $V_{r,v}$ to have a supermajority for $g(H_r)$. By Lemma \ref{lem:ghost-monotonicity}, $g(V_{r,v})$ is on the same chain as $g(H_r)$. For (iii), it is impossible for $V_{r,v}$ to have a supermajority for any children of $g(V_{r,v})$. If we had $g(V_{r,v}) < g(H_r)$, by Lemma \ref{lem:impossible}, this would mean that it would be impossible for $V_{r,v}$ to have a supermajority for $g(H_r)$ as well. So it must be that $g(V_{r,v} )\geq g(H_r)$ as required.

For (b), combining (a) and Lemma \ref{lem:timings} (iii), we have that any honest voter $v$ precommits $\geq g(H_r)$ by time $t_{r,v}+4T$. By our network assumption, all honest participants receive these precommits by time $t_r+6T$ and so finalise $g(H_r)$ if they have not done so already.
\end{proof}

\begin{lemma} \label{lem:primary-finalises}
 Suppose that $t_r \geq GST$, the primary $v$ of round $r$ is honest and no vote has more than $f$ Byzantine voters. Let $B=E_{r-1,v,t_{v,r}}$ be the block $v$ broadcasts if it is not final. Then every honest prevoter prevotes for the best chain including $B$ and all honest voter finalise $B$ by time $t_r+6T$.
 \end{lemma}

\begin{proof} By Lemma \ref{lem:timings} and our network assumptions, no honest voter  prevotes before time $t_r+2T \geq t_{r,v}+2T$ and so at this time, they will have seen all prevotes and precommits seen by $v$ at $t_{r,v}$ and the block $B$ if $v$ broadcast it then. By Lemma \ref{lem:message-monotonicity-completed-estimate}, any honest voter $v'$ has $E_{r-1,v'} \leq B \leq g(V_{r-1,v}$ then.

So if the primary broadcast $B$, then $v'$ prevotes for the best chain including $B$. If the primary did not broadcast $B$, then they finalise it. By Lemma \ref{lem:overestimate-final}, it must be that $E_{r-1,v'} \geq B$ and so $E_{r-1,v'}=B$ and so in this case $v'$ also prevotes for th best chain including $B$.

Since all honest voters prevote $\geq B$, $g(H_r) \geq B$ and so by Lemma \ref{lem:honest-prevote-timings}, all honest participants finalise $B$ by time $t_r+6T$
\end{proof}



\begin{lemma}
 Suppose that $t_r \geq GST+T$ and the primary of round $r$ is honest. 
Let $B$ be the latest block that is ever finalised in rounds  $<r$ (even if no honest participant finalises it until after $t_r$). If all honest voters for the prevote in round $r$ agree that the best chain containing $B$ include the same child $B'$ of $B$, then they all finalises some child of $B$ before $t_r+6T$.
\end{lemma}

\begin{proof} By Lemma \ref{lem:overestimate-final}, any honest participant sees that $E_{r-1} \geq B$ during round $r$. Let $v$ be the primary of round $r$ and $B''=E_{r-1,v,t_{r,v}}$. If $B'' > B$, then by Lemma \ref{lem:primary-finalises}, all honest validators finalise $B''$ by time $t_r+6T$ which means they finalised a child of $B$. If $B''=B$, then by Lemma \ref{lem:honest-prevote-timings}, all honest voters prevote for th best chain including $B$. By assumption these chains include $B'$ and so $g(H_r) \geq B$. By Lemma \ref{lem:honest-prevote-timings}, this means that $B'$ is finalised by time $t_r+6T$.
\end{proof}







\subsubsection{Recent Validity}

\begin{lemma} \label{lem:honest-recent-validity}
Suppose that $t_r \geq GST$, the primary of round $r$ is honest and all votes have at most $f$ Byzantine voters. Let $B$ be a block that no prevoter in round $r$ saw as being in the best chain of an ancestor of $B$ at the time they prevoted. Then either all honest validators finalise $B$ before time $t_r+6T$ or no honest validator ever has $g(V_{r,v}) \geq B$ or $E_{r,v} \geq B$.
\end{lemma}

\begin{proof} Let $v'$ be the primary of round $r$ and let $B'=E_{r-1,v',t_{r,v'}}$. If $B' \geq B$, then by Lemma \ref{lem:primary-finalises}, all honest validators finalise $B$ by time $t_r+6T$. If $B' \not\geq B$, then by Lemma \ref{lem:primary-finalises}, no honest validator prevotes $\geq B$ and so no honest validator ever has $g(V_{r,v}) \geq B$.
\end{proof}


\begin{corollary} For $t - 6T > t' \geq GST$, suppose that an honest validator finalises $B$ at time $t$ but that no honest voter has seen $B$ as in the best chain containing some ancestor of $B$ in between times $t'$ and $t$, then at least $(t-t')/6T - 1$ rounds in a row had Byzantine primaries. \end{corollary}




\section{Practicalities}

\subsection{Changing the voter set on-chain in an asynchronously safe way}

Suppose we have an on-chain protocol that decides we need a different voter set. Once everyone finalises the block, they know that we need to change the set. The protocol can cope with changing the voter set from some round $r$. The main difficulty is that the chain has no idea what the current round number is and even if we have a block that instructs us to change the voter set at round $r$, we might only finalise the block after round $r$. So instead we will not take advantage of the ability to change set from one round to the next. 

A block $B$ can contain an instruction that we should change to the voter set to some other set after some integer $m \geq 0$ blocks. If our best chain for a prevote contains such a block $B$, then we do not prevote for more than $m$ blocks after $B$, even if our best chain is longer. Thus if the current voter set has $n-f$ honest voters, they will only finalise $m$ blocks after such a $B$. We only accept votes and commit messages up top $m$ blocks after $B$ from the current set of validators.  

When some block $B'$ that is $m$ blocks after $B$ has been finalised, then the new validator set starts again at round $1$ with $E_{0}=B'$. Votes will need to contain additional metadata that indicates the validator set somehow.

\subsection{Alternatives to the last block hash}

The danger with voting for the last blockhash in the best chain is that maybe no one else will have seen and processed the next block. It would also be nice to make the most of BLS multisig/aggregation, which allows a single signature for many messages/signers than can be checked in time proportional to the number of different messages signed. 

To get round the first alone, it might be better to vote for a block 3/4 along (rounding further) the unfinalised chain , rather than for th head.

But the second suggests that maybe we should be including signatures for several of the latest blocks in a chain. We could include that last 2 or 3. We could also do e.g. the the blocks with block numbers with the last 2 multiples of each power of two since th last finalised block, which gives log unfinalised chain length messages but should have many blocks in common.

When presented with a vote that includes many blocks, we should interpret them as being for th last block we've seen if any. Then we need to be able to update that vote to a later block when that is seen. This retains monotonicity of a supermajority for/ it is impossible to have a supermajority for over time.

It does not matter if some of the votes are for a block that does not exist as everyone will ignore that part of the vote. But including votes for block that are seen but are not on a chain is an equivocation and is slashable. We need to count such votes as votes for the had of every chain in the vote (as someone might interpret them as for any one of them).

Then if we need to BLS aggregate votes that are $\geq B$ for a commit message or query response, it is OK to use any vote that is $\geq B$, not necessarily the vote for th head. This should reduce the number of blockhashs sign, in the optimistic case down to 1.

\subsection{  Block production rule}

If we adopt that rule that block producers should build on the best chain including the last finalised block, then if we don't finalise another block this will eventually include some prefix  beyond the last finalised block, and therefore the protocol is live by Lemma \ref{lem:honest-recent-validity}.

But the issue is that if agreement is much slower than block production, then we might have a prevote for a short chain on the last finalised block, then the best chain does not include that block and we build a long chain that is eventually never finalised. This could be fixed by building on $E_{r-1}$ or $E_r$. But if we do that, and these change very quickly, then we may never come to agreement on the best chain. 

So we have two possible chain selection rules for block producers:

\begin{enumerate}
\item Build on the best chain including the last finalise block B.
\item Build on best chain including whichever of $\{E_r,E_{r-1},B\}$ is latest and $\geq B$.
\end{enumerate}

1 is better if finalisation is happening quickly compared to block production and 2 is best if block production is much faster. We could also consider hybrid rules like adopt 1 unless we see that the protocol is stuck or slow, then we switch to 2.

\section{Why?}

\subsection{Why do we wait at the end of a round and sometimes before precommitting?}

If the network  is badly behaved, then these steps may involve waiting an arbitrarily long time. When the network is well behaved (after the GST in our model), we should not be waiting. Indeed there is little point not waiting to receive 2/3 of voters' votes as we cannot finalise anything without them. But if the gossip network is not perfect, an some messages never arrive, then we may need to implement voters asking other voters for votes from previous rounds in a similar way to the challenge procedure, to avoid deadlock.

In exchange for this, we get the property that we do not need to pay attention to votes from before the previous round in order to vote correctly in this one. Without waiting, we could be in a situation where we might have finalised a block in some round r, but the network becomes unreliable for many rounds and gets few votes on time, in which case we' need to remember the votes from round r to finalise the block later. 

\subsection{Why have a primary?}

We only need the primary for liveness. We need some form of coordination to defeat the repeated vote splitting attack. The idea behind that attack is that if we are in a situation where almost 2/3 of voters vote for something an the rest vote for another, then the Byzantine voters can control when we see a supermajority for something. If they can carefully time this, they may be able to split the next vote. Without the primary, they could do this for prevotes, getting a supermajority for a block $B$ late, then split precommits so we don't see that it is impossible for there to be a supermajority for $B$ until late. If $B$ is not the best block given the last finalised block but $B'$  with the same block number, they could stop either from being finalised like this even if the (unknown) fraction of Byzantine players is small.

When the network is well-behaved, an honest primary can defeat this attack by deciding how much we should agree on. We could also use a common coin for the same thing, where people would prevote for either the best chain containing $E_{r-1,v}$ or $g(V_{r-1,v})$ depending on the common coin. With on-chain voting, it is possible that we could use probabilistic finality of the block production mechanism - that if we don't finalise a block and always build on the best chain containing the last finalised block then not only will the best chain eventually converge, but if a block is behind the head of the best chain, then with positive probability, it will eventually be in the best chain everyone sees.

In our setup, having a primary is the simplest option for this.
\end{document}