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Restructured from flat .tex/.pdf files into per-paper subdirectories. Added Antje Worring as co-author on all papers. Synced PDFs to pdfs/ for download site.
59 lines
1.9 KiB
TeX
59 lines
1.9 KiB
TeX
\section{State Space Explosion}
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\subsection{The Problem}
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The number of reachable states grows exponentially with the number of components.
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A consensus protocol with $n$ validators, $r$ rounds, and $m$ possible messages has
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roughly $O(n^r \cdot 2^m)$ states. With $n = 7$, $r = 3$, $m = 20$, that is billions
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of states.
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\subsection{Techniques for Managing It}
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\paragraph{Symmetry reduction.}
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If all validators are identical (before assigning Byzantine/honest roles), the model
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checker need not explore permutations separately. TLC supports this via symmetry sets:
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\begin{lstlisting}[language=TLA,caption={Symmetry reduction}]
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\* In TLC configuration:
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\* SYMMETRY Permutations(Validators)
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\end{lstlisting}
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This reduces the state space by a factor of $n!$.
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\paragraph{State constraints.}
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Bound variables to small ranges. Use \texttt{MaxRound = 3} instead of unbounded rounds.
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If a bug exists, it usually manifests in small instances.
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\paragraph{Action constraints.}
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Limit the number of messages in flight:
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\begin{lstlisting}[language=TLA,caption={Action constraint to limit state space}]
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\* In TLC configuration:
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\* ACTION_CONSTRAINT Cardinality(messages) < 20
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\end{lstlisting}
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\paragraph{Abstraction.}
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Replace complex data with simpler representations. Instead of modeling full blocks
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with transactions, use abstract block IDs (\texttt{"B1"}, \texttt{"B2"}).
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\subsection{State Space in Practice}
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\begin{table}[H]
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\centering
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\small
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\begin{tabular}{@{}lrrr@{}}
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\toprule
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\textbf{Configuration} & \textbf{States} & \textbf{Time} & \textbf{RAM} \\
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\midrule
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4 validators, 2 rounds, no symmetry & 12.3M & 4m & 8GB \\
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4 validators, 2 rounds, with symmetry & 0.5M & 11s & 2GB \\
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7 validators, 3 rounds, with symmetry & 84M & 38m & 24GB \\
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7 validators, 3 rounds, no symmetry & $>$10B & timeout & --- \\
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\bottomrule
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\end{tabular}
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\caption{Effect of symmetry reduction on Teleport model checking.}
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\end{table}
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%% ============================================================
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