From 2642c82328cb7d31de8c78128c17cf0b0ae7b41e Mon Sep 17 00:00:00 2001 From: Vitalik Buterin Date: Thu, 17 Aug 2017 18:48:39 -0400 Subject: [PATCH] Some modifications to basic structure paper --- papers/casper/.casper_basic_structure.tex.swp | Bin 36864 -> 0 bytes papers/casper/.casper_economics_basic.tex.swp | Bin 57344 -> 0 bytes papers/casper/casper_basic_structure.log | 4 +-- papers/casper/casper_basic_structure.pdf | Bin 350226 -> 351625 bytes papers/casper/casper_basic_structure.tex | 34 ++++++++++-------- 5 files changed, 21 insertions(+), 17 deletions(-) delete mode 100644 papers/casper/.casper_basic_structure.tex.swp delete mode 100644 papers/casper/.casper_economics_basic.tex.swp diff --git a/papers/casper/.casper_basic_structure.tex.swp b/papers/casper/.casper_basic_structure.tex.swp deleted file mode 100644 index 38cd3c3ec7f8c11e33fa25a5eb30047effc96605..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 36864 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Suppose without loss of generality that $e_A > e_B$. Then, there must be some $e_A^i$ that either $e_A^i = e_B$ or $e_A^i > e_B > e_A^{i+1}$. In the first case, since $A^i$ and $B$ both have $\frac{2}{3}$ prepares, at least $\frac{1}{3}$ of validators violated \textbf{NO\_DBL\_PREPARE}. Otherwise, $B$ has $\frac{2}{3}$ commits and there exist $\frac{2}{3}$ prepares with $epoch > B$ and $epoch_{source} < B$, so at least $\frac{1}{3}$ of validators violated \textbf{PREPARE\_COMMIT\_CONSISTENCY}. This proves safety. +This implies $\frac{2}{3}$ commits and $\frac{2}{3}$ prepares in epochs $e_A$ and $e_B$. In the trivial case where $e_A = e_B$, this implies that some intersection of $\frac{1}{3}$ of validators must have violated \textbf{NO\_DBL\_PREPARE}. In other cases, there must exist two chains $e_A > e_A^1 > e_A^2 > ... > G$ and $e_B > e_B^1 > e_B^2 > ... > G$ of justified checkpoints, both terminating at the genesis. Suppose without loss of generality that $e_A > e_B$. Then, there must be some $e_A^i$ that either $e_A^i = e_B$ or $e_A^i > e_B > e_A^{i+1}$. In the first case, since $A^i$ and $B$ both have $\frac{2}{3}$ prepares, at least $\frac{1}{3}$ of validators violated \textbf{NO\_DBL\_PREPARE}. Otherwise, $B$ has $\frac{2}{3}$ commits and there exist $\frac{2}{3}$ prepares with $epoch > B$ and $epoch_{source} < B$, so at least $\frac{1}{3}$ of validators violated \textbf{PREPARE\_COMMIT\_CONSISTENCY}. This proves accountable safety. -Now, we prove liveness. Suppose that all existing validators have sent some sequence of prepare and commit messages. Let $M$ with epoch $e_M$ be the highest-epoch checkpoint that was justified. Honest validators have not committed on any block which is not justified. Hence, neither slashing condition stops them from making prepares on a child of $M$, using $e_M$ as $epoch_{source}$, and then committing this child. +Now, we prove plausible liveness. Suppose that all existing validators have sent some sequence of prepare and commit messages. Let $M$ with epoch $e_M$ be the highest-epoch checkpoint that was justified. Honest validators have not committed on any block which is not justified. Hence, neither slashing condition stops them from making prepares on a child of $M$, using $e_M$ as $epoch_{source}$, and then committing this child. \section{Fork Choice Rule} @@ -83,7 +87,7 @@ Here is one possible example: \includegraphics[width=400px]{fork_choice_rule.jpeg} -In this case, $HASH1$ or any descendant thereof cannot be finalized without slashing $\frac{1}{6}$ of validators. However, miners on a proof of work chain would interpret $HASH1$ as the head and start mining descendants of it. +In this case, $HASH1$ or any descendant thereof cannot be finalized without slashing $\frac{1}{6}$ of validators. However, miners on a proof of work chain would interpret $HASH1$ as the head and forever keep mining descendants of it, ignoring the chain based on $HASH0'$ which actually could get finalized. In fact, when \textit{any} checkpoint gets $k > \frac{1}{3}$ commits, no conflicting checkpoint can get finalized without $k - \frac{1}{3}$ of validators getting slashed. This necessitates modifying the fork choice rule used by participants in the underlying proposal mechanism (as well as users and validators): instead of blindly following a longest-chain rule, there needs to be an overriding rule that (i) finalized checkpoints are favored, and (ii) when there are no further finalized checkpoints, checkpoints with more (justified) commits are favored. @@ -96,9 +100,9 @@ One complete description of such a rule would be: \item Choose the longest proof of work chain from there. \end{enumerate} -The commit-following part of this rule can be viewed in some ways as mirroring the "greegy heaviest observed subtree" (GHOST) rule that has been proposed for proof of work chains [cite]. The symmetry is this: in GHOST, a node starts with the head at the genesis, then begins to move forward down the chain, and if it encounters a block with multiple children then it chooses the child that has the larger quantity of work built on top of it (including the child block itself and its descendants). +The commit-following part of this rule can be viewed in some ways as mirroring the "greegy heaviest observed subtree" (GHOST) rule that has been proposed for proof of work chains [cite]. The symmetry is as follows. In GHOST, a node starts with the head at the genesis, then begins to move forward down the chain, and if it encounters a block with multiple children then it chooses the child that has the larger quantity of work built on top of it (including the child block itself and its descendants). -Here, we follow a similar approach, except we repeatedly seek the child that comes the closest to achieving finality. Commits on a descendant are implicitly commits on all of its ancestors, and so if a given descendant of a given block has more commits than any other descendant, then we know that all children along the chain from the head to this descendant are closer to finality than any of their siblings; hence, looking for the \textit{descendant} with the most commits and not just the \textit{child} replicates the GHOST principle most faithfully. Finalizing a checkpoint requires 2/3 commits within a single epoch, and so we do not try to sum up commits across epochs and instead simply take the maximum. +In this algorithm, we follow a similar approach, except we repeatedly seek the child that comes the closest to achieving finality. Commits on a descendant are implicitly commits on all of its ancestors, and so if a given descendant of a given block has more commits than any other descendant, then we know that all children along the chain from the head to this descendant are closer to finality than any of their siblings; hence, looking for the \textit{descendant} with the most commits and not just the \textit{child} replicates the GHOST principle most faithfully. Finalizing a checkpoint requires $\frac{2}{3}$ commits within a \textit{single} epoch, and so we do not try to sum up commits across epochs and instead simply take the maximum. This rule ensures that if there is a checkpoint such that no conflicting checkpoint can be finalized without at least some validators violating slashing conditions, then this is the checkpoint that will be viewed as the ``head'' and thus that validators will try to commit on.