From 28b6d42faea6ad994be00c6d4fae0e663bca0b60 Mon Sep 17 00:00:00 2001 From: Vitalik Buterin Date: Tue, 15 Aug 2017 04:11:54 -0400 Subject: [PATCH] Changes to basic structure paper --- papers/casper/casper_basic_structure.aux | 19 ++++---- papers/casper/casper_basic_structure.log | 57 ++++++++++++----------- papers/casper/casper_basic_structure.pdf | Bin 178027 -> 185707 bytes papers/casper/casper_basic_structure.tex | 48 +++++++++++++++---- 4 files changed, 81 insertions(+), 43 deletions(-) diff --git a/papers/casper/casper_basic_structure.aux b/papers/casper/casper_basic_structure.aux index 02f75bb..f472203 100644 --- a/papers/casper/casper_basic_structure.aux +++ b/papers/casper/casper_basic_structure.aux @@ -1,12 +1,13 @@ \relax -\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}} -\@writefile{toc}{\contentsline {section}{\numberline {2}Rewards and Penalties}{2}} -\@writefile{toc}{\contentsline {section}{\numberline {3}Claims}{4}} -\@writefile{toc}{\contentsline {section}{\numberline {4}Individual choice analysis}{5}} -\@writefile{toc}{\contentsline {section}{\numberline {5}Collective choice model}{5}} -\@writefile{toc}{\contentsline {section}{\numberline {6}Griefing factor analysis}{7}} -\@writefile{toc}{\contentsline {section}{\numberline {7}Pools}{9}} +\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction, Protocol I}{1}} +\@writefile{toc}{\contentsline {section}{\numberline {2}Proof Sketch of Safety and Plausible Liveness}{3}} +\@writefile{toc}{\contentsline {section}{\numberline {3}Dynamic Validator Sets}{4}} +\@writefile{toc}{\contentsline {section}{\numberline {4}Claims}{6}} +\@writefile{toc}{\contentsline {section}{\numberline {5}Individual choice analysis}{6}} +\@writefile{toc}{\contentsline {section}{\numberline {6}Collective choice model}{7}} +\@writefile{toc}{\contentsline {section}{\numberline {7}Griefing factor analysis}{9}} +\@writefile{toc}{\contentsline {section}{\numberline {8}Pools}{11}} \bibstyle{abbrv} \bibdata{main} -\@writefile{toc}{\contentsline {section}{\numberline {8}Conclusions}{11}} -\@writefile{toc}{\contentsline {section}{\numberline {9}References}{11}} +\@writefile{toc}{\contentsline {section}{\numberline {9}Conclusions}{13}} +\@writefile{toc}{\contentsline {section}{\numberline {10}References}{13}} diff --git a/papers/casper/casper_basic_structure.log b/papers/casper/casper_basic_structure.log index ebf7827..6684ed5 100644 --- a/papers/casper/casper_basic_structure.log +++ b/papers/casper/casper_basic_structure.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.6.27) 7 AUG 2017 03:41 +This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.6.27) 15 AUG 2017 04:11 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -76,7 +76,7 @@ Package: array 2014/10/28 v2.4c Tabular extension package (FMi) ) \c@definition=\count89 -No file casper_basic_structure.aux. +(./casper_basic_structure.aux) \openout1 = `casper_basic_structure.aux'. 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conditions. + +\section{Proof Sketch of Safety and Plausible Liveness} + +We give a proof sketch of two properties of this scheme: \textit{safety} and \textit{plausible liveness}. Safety means that two incompatible checkpoints cannot be finalized unless at least $\frac{1}{3}$ of validators violate a slashing condition. Plausible liveness means that it is always possible for $\frac{2}{3}$ of honest validators to finalize a new checkpoint, regardless of what previous events took place. + +Suppose that two incompatible checkpoints $A$ (epoch $e_A$) and $B$ (epoch $e_B$) are finalized: + +[diagram] + +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 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. + +\section{Dynamic Validator Sets} We define the following constants and functions: