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color code levenshtein automaton
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@breandan
breandan committed Feb 27, 2024
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205 changes: 97 additions & 108 deletions latex/splash2024/nfa_cfg.tex
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]
\node[state, initial] (00) {$q_{0,0}$};
\node[state, right of=00] (10) {$q_{1,0}$};
\node[state, right of=10] (20) {$q_{2,0}$};
\node[state, right of=20] (30) {$q_{3,0}$};
\node[right of=30] (40) {$\vphantom{\vdots}\cdots$};
\node[accepting, state, right of=40] (n0) {$q_{n,0}$};

\node[state, above of=00, shift={(-0.7cm,-0.0cm)}] (01) {$q_{0,1}$};
\node[state, right of=01] (11) {$q_{1,1}$};
\node[state, right of=11] (21) {$q_{2,1}$};
\node[state, right of=21] (31) {$q_{3,1}$};
\node[right of=31] (41) {$\vphantom{\vdots}\cdots$};
\node[accepting, state, right of=41] (n1) {$q_{n,1}$};

\node[above of=01, shift={(-0.7cm,-0.0cm)}] (0j) {$\mathmakebox[\widthof{$\cdots$}]{\vdots}$};
\node[right of=0j] (1j) {$\mathmakebox[\widthof{$\cdots$}]{\vdots}$};
\node[right of=1j] (2j) {$\mathmakebox[\widthof{$\cdots$}]{\vdots}$};
\node[right of=2j] (3j) {$\mathmakebox[\widthof{$\cdots$}]{\vdots}$};
\node[right of=3j] (4j) {$\iddots$};
\node[accepting, right of=4j] (nj) {$\mathmakebox[\widthof{$\cdots$}]{\vdots}$};

\node[state, above of=0j, shift={(-0.7cm,0.0cm)}] (0k) {$q_{0,k}$};
\node[state, right of=0k] (1k) {$q_{1,k}$};
\node[state, right of=1k] (2k) {$q_{2,k}$};
\node[state, right of=2k] (3k) {$q_{3,k}$};
\node[right of=3k] (4k) {$\vphantom{\vdots}\cdots$};
\node[accepting, state, right of=4k] (nk) {$q_{n,k}$};
\node[accepting, state, right of=10] (20) {$q_{2,0}$};
\node[accepting, state, right of=20] (30) {$q_{3,0}$};
\node[accepting, state, right of=30] (40) {$q_{4,0}$};
\node[accepting, state, right of=40] (50) {$q_{5,0}$};

\node[state, above of=00, shift={(-1.5cm,-0.0cm)}] (01) {$q_{0,1}$};
\node[state, right of=01] (11) {$q_{1,1}$};
\node[state, right of=11] (21) {$q_{2,1}$};
\node[accepting, state, right of=21] (31) {$q_{3,1}$};
\node[accepting, state, right of=31] (41) {$q_{4,1}$};
\node[accepting, state, right of=41] (51) {$q_{5,1}$};

\node[state, above of=01, shift={(-1.5cm,-0.0cm)}] (0j) {$q_{0,2}$};
\node[state, right of=0j] (1j) {$q_{1,2}$};
\node[state, right of=1j] (2j) {$q_{2,2}$};
\node[state, right of=2j] (3j) {$q_{3,2}$};
\node[accepting, state, right of=3j] (4j) {$q_{4,2}$};
\node[accepting, state, right of=4j] (5j) {$q_{5,2}$};

\node[state, above of=0j, shift={(-1.5cm,0.0cm)}] (0k) {$q_{0,3}$};
\node[state, right of=0k] (1k) {$q_{1,3}$};
\node[state, right of=1k] (2k) {$q_{2,3}$};
\node[state, right of=2k] (3k) {$q_{3,3}$};
\node[state, right of=3k] (4k) {$q_{4,3}$};
\node[accepting, state, right of=4k] (5k) {$q_{5,3}$};

\draw [->] (00) edge[below] node{$\sigma_1$} (10);
\draw [->] (10) edge[below] node{$\sigma_2$} (20);
\draw [->] (20) edge[below] node{$\sigma_3$} (30);
\draw [->] (30) edge[below] node{$\sigma_4$} (40);
\draw [->] (40) edge[below] node{$\sigma_n$} (n0);
\draw [->] (40) edge[below] node{$\sigma_5$} (50);

\draw [->] (01) edge[below] node{$\sigma_1$} (11);
\draw [->] (11) edge[below] node{$\sigma_2$} (21);
\draw [->] (21) edge[below] node{$\sigma_3$} (31);
\draw [->] (31) edge[below] node{$\sigma_4$} (41);
\draw [->] (41) edge[below] node{$\sigma_n$} (n1);
\draw [->] (41) edge[below] node{$\sigma_5$} (51);

\draw [->] (0j) edge[below] node{$\sigma_1$} (1j);
\draw [->] (1j) edge[below] node{$\sigma_2$} (2j);
\draw [->] (2j) edge[below] node{$\sigma_3$} (3j);
\draw [->] (3j) edge[below] node{$\sigma_4$} (4j);
\draw [->] (4j) edge[below] node{$\sigma_n$} (nj);
\draw [->] (4j) edge[below] node{$\sigma_5$} (5j);

\draw [->] (0k) edge[below] node{$\sigma_1$} (1k);
\draw [->] (1k) edge[below] node{$\sigma_2$} (2k);
\draw [->] (2k) edge[below] node{$\sigma_3$} (3k);
\draw [->] (3k) edge[below] node{$\sigma_4$} (4k);
\draw [->] (4k) edge[below] node{$\sigma_n$} (nk);

\draw [->] (00) edge[left] node{$\cdot$} (11);
\draw [->] (10) edge[left] node{$\cdot$} (21);
\draw [->] (20) edge[left] node{$\cdot$} (31);
\draw [->] (30) edge[left] node{$\cdot$} (41);
\draw [->, blue] (30) edge[bend right, below] node{$\color{blue}\sigma_5$} (41);
\draw [->] (40) edge[ right] node{$\sigma_n$} (n1);
\draw [->] (40) edge[bend right, left] node{$\cdot$} (n1);

\draw [->, red] (00) edge[bend left=2] node[east, shift={(-0.5cm,-0.8cm)}]{$\color{red}\sigma_3$} (3j);
\draw [->, red] (10) edge[bend left=2] node[east, shift={(-0.5cm,-0.8cm)}]{$\color{red}\sigma_4$} (4j);
%\draw [->, red] (20) edge[bend left=7] node[east, shift={(-0.7cm,-0.8cm)}]{$\color{red}\sigma_5$} (3j);
%\draw [->, red] (30) edge[bend left=7] node[east, shift={(-0.7cm,-0.8cm)}]{$\color{red}\sigma_6$} (4j);

\draw [->] (01) edge[left] node{$\cdot$} (1j);
\draw [->] (11) edge[left] node{$\cdot$} (2j);
\draw [->] (21) edge[left] node{$\cdot$} (3j);
\draw [->] (31) edge[left] node{$\cdot$} (4j);
\draw [->, blue] (31) edge[bend right, below] node{$\color{blue}\sigma_5$} (4j);
\draw [->] (41) edge[ right] node{$\sigma_n$} (nj);
\draw [->] (41) edge[bend right, left] node{$\cdot$} (nj);

\draw [->] (0j) edge[left] node{$\cdot$} (1k);
\draw [->] (1j) edge[left] node{$\cdot$} (2k);
\draw [->] (2j) edge[left] node{$\cdot$} (3k);
\draw [->] (3j) edge[left] node{$\cdot$} (4k);
\draw [->, blue] (3j) edge[bend right, below] node{$\color{blue}\sigma_5$} (4k);
\draw [->] (4j) edge[ right] node{$\sigma_n$} (nk);
\draw [->] (4j) edge[bend right, left] node{$\cdot$} (nk);

\draw [->] (00) edge[bend left, left] node{$\cdot$} (01);
\draw [->] (10) edge[bend left, left] node{$\cdot$} (11);
\draw [->] (20) edge[bend left, left] node{$\cdot$} (21);
\draw [->] (30) edge[bend left, left] node{$\cdot$} (31);
\draw [->] (40) edge[right] node{$\cdot$} (41);
\draw [->] (n0) edge[bend right, right] node{$\cdot$} (n1);

\draw [->] (01) edge[bend left, left] node{$\cdot$} (0j);
\draw [->] (11) edge[bend left, left] node{$\cdot$} (1j);
\draw [->] (21) edge[bend left, left] node{$\cdot$} (2j);
\draw [->] (31) edge[bend left, left] node{$\cdot$} (3j);
\draw [->] (41) edge[right] node{$\cdot$} (4j);
\draw [->] (n1) edge[bend right, right] node{$\cdot$} (nj);

\draw [->] (0j) edge[bend left, left] node{$\cdot$} (0k);
\draw [->] (1j) edge[bend left, left] node{$\cdot$} (1k);
\draw [->] (2j) edge[bend left, left] node{$\cdot$} (2k);
\draw [->] (3j) edge[bend left, left] node{$\cdot$} (3k);
\draw [->] (4j) edge[right] node{$\cdot$} (4k);
\draw [->] (nj) edge[bend right, right] node{$\cdot$} (nk);

\draw [->, blue] (00) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_2$} (21);
\draw [->, blue] (10) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_3$} (31);
\draw [->, blue] (20) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (41);

\draw [->, blue] (01) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_2$} (2j);
\draw [->, blue] (11) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_3$} (3j);
\draw [->, blue] (21) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (4j);

\draw [->, blue] (0j) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_2$} (2k);
\draw [->, blue] (1j) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_3$} (3k);
\draw [->, blue] (2j) edge[below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (4k);
\draw [->] (4k) edge[below] node{$\sigma_5$} (5k);

\draw [->] (00) edge[left] node{$\phantom{\cdot}$} (11);
\draw [->] (10) edge[left] node{$\phantom{\cdot}$} (21);
\draw [->] (20) edge[left] node{$\phantom{\cdot}$} (31);
\draw [->] (30) edge[left] node{$\phantom{\cdot}$} (41);
\draw [->] (40) edge[left] node{$\phantom{\cdot}$} (51);

% Super-knight arcs
\draw [->, red] (00) edge[bend left=3] node[east, shift={(-0.3cm,-0.65cm)}]{$\color{red}\sigma_3$} (3j);
\draw [->, red] (10) edge[bend left=3] node[east, shift={(-0.3cm,-0.65cm)}]{$\color{red}\sigma_4$} (4j);
\draw [->, red] (20) edge[bend left=3] node[east, shift={(-0.3cm,-0.65cm)}]{$\color{red}\sigma_5$} (5j);
\draw [->, red] (01) edge[bend left=3] node[east, shift={(-0.3cm,-0.65cm)}]{$\color{red}\sigma_3$} (3k);
\draw [->, red] (11) edge[bend left=3] node[east, shift={(-0.3cm,-0.65cm)}]{$\color{red}\sigma_4$} (4k);
\draw [->, red] (21) edge[bend left=3] node[east, shift={(-0.3cm,-0.65cm)}]{$\color{red}\sigma_5$} (5k);

\draw [->, violet] (00) edge[bend right=1] node[east, shift={(0cm,0.35cm)}]{$\color{violet}\sigma_4$} (4k);
\draw [->, violet] (10) edge[bend right=1] node[east, shift={(0cm,0.35cm)}]{$\color{violet}\sigma_5$} (5k);

\draw [->] (01) edge[left] node{$\phantom{\cdot}$} (1j);
\draw [->] (11) edge[left] node{$\phantom{\cdot}$} (2j);
\draw [->] (21) edge[left] node{$\phantom{\cdot}$} (3j);
\draw [->] (31) edge[left] node{$\phantom{\cdot}$} (4j);
\draw [->] (41) edge[left] node{$\phantom{\cdot}$} (5j);

\draw [->] (0j) edge[left] node{$\phantom{\cdot}$} (1k);
\draw [->] (1j) edge[left] node{$\phantom{\cdot}$} (2k);
\draw [->] (2j) edge[left] node{$\phantom{\cdot}$} (3k);
\draw [->] (3j) edge[left] node{$\phantom{\cdot}$} (4k);
\draw [->] (4j) edge[left] node{$\phantom{\cdot}$} (5k);

\draw [->] (00) edge[bend left=10, left] node{$\phantom{\cdot}$} (01);
\draw [->] (10) edge[bend left=10, left] node{$\phantom{\cdot}$} (11);
\draw [->] (20) edge[bend left=10, left] node{$\phantom{\cdot}$} (21);
\draw [->] (30) edge[bend left=10, left] node{$\phantom{\cdot}$} (31);
\draw [->] (40) edge[bend left=10, left] node{$\phantom{\cdot}$} (41);
\draw [->] (50) edge[bend left=10, left] node{$\phantom{\cdot}$} (51);

\draw [->] (01) edge[bend left=10, left] node{$\phantom{\cdot}$} (0j);
\draw [->] (11) edge[bend left=10, left] node{$\phantom{\cdot}$} (1j);
\draw [->] (21) edge[bend left=10, left] node{$\phantom{\cdot}$} (2j);
\draw [->] (31) edge[bend left=10, left] node{$\phantom{\cdot}$} (3j);
\draw [->] (41) edge[bend left=10, left] node{$\phantom{\cdot}$} (4j);
\draw [->] (51) edge[bend left=10, left] node{$\phantom{\cdot}$} (5j);

\draw [->] (0j) edge[bend left=10, left] node{$\phantom{\cdot}$} (0k);
\draw [->] (1j) edge[bend left=10, left] node{$\phantom{\cdot}$} (1k);
\draw [->] (2j) edge[bend left=10, left] node{$\phantom{\cdot}$} (2k);
\draw [->] (3j) edge[bend left=10, left] node{$\phantom{\cdot}$} (3k);
\draw [->] (4j) edge[bend left=10, left] node{$\phantom{\cdot}$} (4k);
\draw [->] (5j) edge[bend left=10, left] node{$\phantom{\cdot}$} (5k);

\draw [->, blue] (00) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_2$} (21);
\draw [->, blue] (10) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_3$} (31);
\draw [->, blue] (20) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (41);
\draw [->, blue] (30) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_5$} (51);

\draw [->, blue] (01) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_2$} (2j);
\draw [->, blue] (11) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_3$} (3j);
\draw [->, blue] (21) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (4j);
\draw [->, blue] (31) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (5j);

\draw [->, blue] (0j) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_2$} (2k);
\draw [->, blue] (1j) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_3$} (3k);
\draw [->, blue] (2j) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_4$} (4k);
\draw [->, blue] (3j) edge[bend left=2,below] node[shift={(-0.7cm,-0.5cm)}]{$\color{blue}\sigma_5$} (5k);

%https://tex.stackexchange.com/a/20986/139648
\draw [decorate,decoration={brace,amplitude=10pt,raise=10pt,mirror}] (00.south west) -- (n0.south east) node[midway,yshift=-3em]{\textbf{String length}};
\draw [decorate,decoration={brace,amplitude=10pt,raise=20pt}] (00.south west) -- (0k.north west) node[midway,xshift=-40pt,rotate=-75]{\textbf{Edit distance}};
\draw [decorate,decoration={brace,amplitude=10pt,raise=10pt,mirror}] (00.south west) -- (50.south east) node[midway,yshift=-3em]{\textbf{String length}};
\draw [decorate,decoration={brace,amplitude=10pt,raise=20pt}] (00.south west) -- (0k.north west) node[midway,xshift=-1.2cm,yshift=-0.6cm,rotate=-60]{\textbf{Edit distance}};
\end{tikzpicture}
}
%\end{minipage}
%\hfill
%\begin{minipage}[l]{5 cm}
%\centering
%\underline{CFG}\vspace{2pt}
%\begin{align*}
%S &\Rightarrow \{\cdot \in Q \mid \delta(\cdot, q_{n,0}) \leq k\}\\
%+ &\Rightarrow \{\cdot \in \Sigma\}\\
%\big\{q_{i, j} &\Rightarrow \{q_{i, j-1}+\} \mid i, j \in [0, n]\times[1, k]\big\}\\
%\big\{q_{i, j} &\Rightarrow \{q_{i-1, j-1}+\}\mid i, j\in[1, n]\times [1, k]\big\}\\
%\big\{q_{i, j} &\Rightarrow \{q_{i-1, j} \sigma_i \}\mid i, j \in [1, n]\times[0, k]\big\} \\
%\big\{q_{i, j} &\Rightarrow \{q_{i-2, j-1} \sigma_i\} \mid i, j \in [2, n]\times[1, k] \big\}\\
%\end{align*}
%\end{minipage}
}
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10 changes: 5 additions & 5 deletions latex/splash2024/splash.tex
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Expand Up @@ -346,7 +346,7 @@
\begin{center}
\resizebox{.9\textwidth}{!}{\input{nfa_cfg.tex}}
\end{center}
\caption{Levenshtein reachability from $\Sigma^n$ can be described as an NFA, or a CFG.}
\caption{Automaton recognizing Levenshtein $\Delta(\sigma: \Sigma^5, 3)$ reachability. Unlabeled arcs accept any terminal.}
\end{figure}

\noindent Alternatively, this transition system can be viewed as a kind of proof system.
Expand All @@ -362,14 +362,14 @@
\UnaryInfC{$(q_{i-1, j-1} \overset{s}{\rightarrow} q_{i,j}) \in \delta$}
\end{prooftree}
\begin{prooftree}
\AxiomC{$s=\sigma_i \phantom{\land} i \in [1, n] \phantom{\land} j \in [0, k]$}
\AxiomC{$i \in [1, n] \phantom{\land} j \in [0, k]$}
\RightLabel{$\drightarrow$}
\UnaryInfC{$(q_{i-1, j} \overset{s}{\rightarrow} q_{i,j}) \in \delta$}
\UnaryInfC{$(q_{i-1, j} \overset{\sigma_i}{\rightarrow} q_{i,j}) \in \delta$}
\DisplayProof
\hskip 1.5em
\AxiomC{$s=\sigma_i \phantom{\land} i \in [2, n] \phantom{\land} j \in [1, k]$}
\AxiomC{$d \in [1, d_{\max}] \phantom{\land} i \in [d + 1, n] \phantom{\land} j \in [d, k]$}
\RightLabel{$\knightarrow$}
\UnaryInfC{$(q_{i-2, j-1} \overset{s}{\rightarrow} q_{i,j}) \in \delta$}
\UnaryInfC{$(q_{i-d-1, j-d} \overset{\sigma_i}{\rightarrow} q_{i,j}) \in \delta$}
\end{prooftree}
\begin{prooftree}
\AxiomC{$\vphantom{|}$}
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