expand "see"

pdf/grandpa.tex

@@ -70,7 +70,7 @@ We can change this definition to assume that instead of having an initial value,
 In the case where $|S| > 2$, this definition of validity is stronger than the validity notion in multi-valued Byzantine agreement ported here verbatim because all honest voters decide a value with which some honest voter started. 
 This is because our earlier definition would be impossible if the fraction of Byzantine voters exceeds $1/|S|$, as we cannot detect Byzantine voters who act like honest voters, except for lying about their initial value. So if fewer than $1/|S|$ voters act like they have some initial value, the protocol cannot know if any are honest. 
 
-But for the case $|S|=2$, the two possible definitions of validity are equivalent. This means that we can reduce the binary version of the Byzantine finality gadget problem above to binary Byzantine agreement, by each voter just calling $A$ at the start to obtain their initial value, since if $A$ does not return the same value to every honest voter all the time, then it returns both values to honest voters some times. Thus there are many existing algorithms for the binary Byzantine finality gadget problem. However the interesting problem in this case is whether the celebrated impossibility result of \cite{flp} generalizes to this finality gadget problem, i.e., whether this oracle which is guaranteed to achieve eventual consensus makes it possible to have an asynchronous and deterministic protocol for agreement. A reduction is not immediately obvious. It turns out that the finality gadget version is indeed impossible see \ref{ssec:impossibility}.
+But for the case $|S|=2$, the two possible definitions of validity are equivalent. This means that we can reduce the binary version of the Byzantine finality gadget problem above to binary Byzantine agreement, by each voter just calling $A$ at the start to obtain their initial value, since if $A$ does not return the same value to every honest voter all the time, then it returns both values to honest voters some times. Thus there are many existing algorithms for the binary Byzantine finality gadget problem. However the interesting problem in this case is whether the celebrated impossibility result of \cite{flp} generalizes to this finality gadget problem, i.e., whether this oracle which is guaranteed to achieve eventual consensus makes it possible to have an asynchronous and deterministic protocol for agreement. A reduction is not immediately obvious. It turns out that the finality gadget version is indeed impossible, as we shall see in \ref{ssec:impossibility}.
 
 Now how do we extend this to agreeing on a chain of blocks? We will need the block production mechanism to build on finalised blocks, so the best chain rule must include them. We assume a kind of conditional eventual consensus. If we keep building on our last finalised block $B$ and don't finalise any new blocks, then eventually we have consensus on a longer chain than just $B$, which the finality gadget can use to finalise another block. We also want a protocol that does not terminate, but instead keeps on finalising more blocks.