Paper: dropped explicit submatch indices in TNFA definition.

diff --git a/re2c/doc/tdfa_v2/img/tnfa.tex b/re2c/doc/tdfa_v2/img/tnfa.tex
index 6a8ebb00aa1e5d29fceca3c814bb617cbddbbd57..c800a0715929183567a167aa8e181bfdcad04159 100644 (file)
--- a/re2c/doc/tdfa_v2/img/tnfa.tex
+++ b/re2c/doc/tdfa_v2/img/tnfa.tex
@@ -27,27 +27,51 @@
 \tikzset{style1/.style={draw, rectangle, rounded corners = 10, minimum width = \widd, minimum height = 0.3in, xshift = \offs}}
 \tikzset{style2/.style={state, accepting, xshift = \offs}}
 
+\begin{scope}[xshift=3.3in, yshift=-7.2in]
+    \node[draw=none, shape=rectangle] (x) {$
+    \begin{aligned}
+    otag \big( (i, j, r) \big) =
+        \begin{cases}
+            \epsilon &\text{if } i = 0 \\[-0.5em]
+            2i - 1   &\text{if } i \neq 0
+        \end{cases}
+    \qquad\qquad
+    ctag \big( (i, j, r) \big) =
+        \begin{cases}
+            \epsilon &\text{if } i = 0 \\[-0.5em]
+            2i       &\text{if } i \neq 0
+        \end{cases}
+    \qquad\qquad
+    ntag \big( (i, j, r) \big) =
+        \begin{cases}
+            \epsilon  &\text{if } i = 0 \\[-0.5em]
+            1 - 2i    &\text{if } i \neq 0
+        \end{cases}
+    \end{aligned}
+$};
+\end{scope}
+
 \begin{scope}[xshift=0in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
     \node[state] (a) {$x$};
     \node[state, accepting, right of=a] (b1) {$x_1$};
-    \node[style1, right of = b1] (b) {$F \big( (0, 0, r) \big)$};
+    \node[style1, right of = b1] (b) {$F \big( (0, 0, r_1) \big)$};
     \node[style2, right of = b] (b2) {$y_1$};
     \node[state, accepting, right of=b2] (d) {$y$};
     \path
-        (a) edge node {$\epsilon / otag(r)$} (b1)
-        (b2) edge node {$\epsilon / ctag(r)$} (d)
+        (a) edge node {$1 / otag(r) $} (b1)
+        (b2) edge node {$1 / ctag(r) $} (d)
     ;
     \path (a) edge [draw=none] node [below=0.15in, midway] {
     $\begin{aligned}
-    F \big( (i, j, r) \big) \mid i \neq 0 &= (\Sigma, Q, x, \{y\}, \Delta) \\
+    F \big( r \big) \mid_{r \;=\; (i, \Xund, r_1) \;\wedge\; i \;\neq\; 0} &= (\Sigma, Q, x, \{y\}, \Delta) \\
         \text{where }
-            F \big( (0, 0, r) \big) &= (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1) \\
+            F \big( (0, 0, r_1) \big) &= (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1) \\
             Q &= Q_1 \cup \{ x, y \}\\
             \Delta &= \Delta_1 \cup
-                \big\{ (x, \epsilon, otag(r), x_1), (y_1, \epsilon, ctag(r), y) \big\}
+                \big\{ (x, 1, otag(r), x_1), (y_1, 1, ctag(r), y) \big\}
     \end{aligned}$
     } (d);
 \end{scope}
@@ -55,14 +79,14 @@
 \begin{scope}[xshift=0.5in, yshift=-1.4in]
     \node[state] (a) {$x$};
     \node[state, accepting, right of=a] (b) {$y$};
-    \path (a) edge node {$\epsilon / \epsilon$} (b);
+    \path (a) edge node {$1 / \epsilon$} (b);
     \path (a) edge [draw=none] node [below=0.15in, midway] {
     $F \big( (0, 0, \epsilon) \big) = \big(
         \Sigma,
         \{x, y\},
         x,
         \{y\},
-        \{ (x, \epsilon, \epsilon, y) \}
+        \{ (x, 1, \epsilon, y) \}
     \big)$
     } (b);
 \end{scope}
@@ -115,10 +139,10 @@
 %    \node[draw, rectangle, rounded corners = 6, minimum width = 0.17in, minimum height = 0.67in, right of = b, xshift = \offs, yshift = -0.25in] (ab) {};
     \node[state, accepting] (d) [below right of = b2, yshift = 0.4in] {$y$};
     \path
-        (a)  edge [bend left]  node {$\epsilon / \epsilon$} (b1)
-        (a)  edge [bend right] node [below left] {$\epsilon / ntag(r_1) $} (c1)
-        (b2) edge [bend left] node {$\epsilon / ntag(r_2) $} (d)
-        (c2) edge [bend right] node [below right] {$\epsilon / \epsilon$} (d)
+        (a)  edge [bend left]  node {$1 / \epsilon$} (b1)
+        (a)  edge [bend right] node [below left] {$2 / ntag(r_1) $} (c1)
+        (b2) edge [bend left] node {$1 / ntag(r_2) $} (d)
+        (c2) edge [bend right] node [below right] {$1 / \epsilon$} (d)
     ;
 %    \path (a) edge [draw=none] node [below=0.6in, midway] {$F \big( (0, 0, (i_1, j_1, r_1) \mid (i_2, j_2, r_2)) \big)$} (d);
     \path (a) edge [draw=none] node [below=0.6in, midway] {
@@ -128,8 +152,8 @@
             F(r_i) &= (\Sigma, Q_i, x_i, \{y_i\}, \Delta_i) \; \forall i = \overline{1,2} \\
             Q &= Q_1 \cup Q_2 \cup \{ x, y \} \\
             \Delta &= \Delta_1 \cup \Delta_2 \cup
-                \big\{ (x, \epsilon, \epsilon, x_1), (y_2, \epsilon, \epsilon, y), \\
-                    & (y_1, \epsilon, ntag(r_2), y), (x, \epsilon, ntag(r_1), x_2)
+                \big\{ (x, 1, \epsilon, x_1), (y_2, 1, \epsilon, y), \\
+                    & (y_1, 1, ntag(r_2), y), (x, 2, ntag(r_1), x_2)
             \big\}
     \end{aligned}$} (d);
 \end{scope}
@@ -145,7 +169,7 @@
     \node[style2, right of = b, xshift = 0.4in] (b2) {};
     \path (a1) edge [draw=none] node [below=0.25in, midway] {
     $\begin{aligned}
-    F \big( (0, 0, r^{n, m}) \big) \mid n > 1 &= F \big( (0, 0, r \cdot (0, 0, r^{n-1, m-1})) \big)
+    F \big( (0, 0, r^{n, m}) \big) \mid_{n \;>\; 1} &= F \big( (0, 0, r \cdot (0, 0, r^{n-1, m-1})) \big)
     \end{aligned}$} (b2);
 \end{scope}
 
@@ -161,9 +185,9 @@
     \node[style2, right of = b] (b2) {$y_1$};
 
     \path
-        (a) edge node {$\epsilon / \epsilon$} (b1)
+        (a) edge node {$1 / \epsilon$} (b1)
     ;
-    \draw (a) .. controls ($ (a) + (0, 1.5) $) and ($ (b2) + (-1, 1) $) .. node [above] {$\epsilon / ntag(r)$} (b2);
+    \draw (a) .. controls ($ (a) + (0, 1.5) $) and ($ (b2) + (-1, 1) $) .. node [above] {$2 / ntag(r)$} (b2);
     \path (a) edge [draw=none] node [below=0.3in, midway] {
     $\begin{aligned}
     F \big( (0, 0, r^{0, m}) \big) &= (\Sigma, Q, x, \{y_1\}, \Delta) \\
@@ -171,7 +195,7 @@
             F \big( (0, 0, r^{1, m}) \big) &= (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1) \\
             Q &= Q_1 \cup \{ x \} \\
             \Delta &= \Delta_1 \cup
-                \big\{ (x, \epsilon, \epsilon, x_1), (x, \epsilon, ntag(r), y_1) \big\}
+                \big\{ (x, 1, \epsilon, x_1), (x, 2, ntag(r), y_1) \big\}
     \end{aligned}$} (b2);
 \end{scope}
 
@@ -187,9 +211,9 @@
     \node[state, accepting, right of = b2] (c) {$y$};
 
     \path
-        (b2) edge node {$\epsilon / \epsilon$} (c)
+        (b2) edge node {$1 / \epsilon$} (c)
     ;
-    \draw (b2) .. controls ($ (b2) + (1, 1.5) $) and ($ (b1) + (-1, 1.5) $) .. node [above] {$\epsilon / \epsilon$} (b1);
+    \draw (b2) .. controls ($ (b2) + (1, 1.5) $) and ($ (b1) + (-1, 1.5) $) .. node [above] {$2 / \epsilon$} (b1);
     \path (b1) edge [draw=none] node [below=0.3in, midway] {
     $\begin{aligned}
     F \big( (0, 0, r^{1, \infty}) \big) &= (\Sigma, Q, x_1, \{y\}, \Delta) \\
@@ -197,7 +221,7 @@
             F(r) &= (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1) \\
             Q &= Q_1 \cup \{ y \} \\
             \Delta &= \Delta_1 \cup
-                \big\{ (y_1, \epsilon, \epsilon, x_1), (y_1, \epsilon, \epsilon, y) \big\}
+                \big\{ (y_1, 2, \epsilon, x_1), (y_1, 1, \epsilon, y) \big\}
     \end{aligned}$} (c);
 \end{scope}
 
@@ -221,12 +245,12 @@
     \node[style2, right of = d] (d2) {$y_m$};
 
     \path
-        (c2) edge node {$\epsilon / \epsilon$} (cd)
-        (cd) edge node {$\epsilon / \epsilon$} (d1)
-        (b2) edge node {$\epsilon / \epsilon$} (c1)
+        (c2) edge node {$2 / \epsilon$} (cd)
+        (cd) edge node {$2 / \epsilon$} (d1)
+        (b2) edge node {$2 / \epsilon$} (c1)
     ;
-    \draw (b2) .. controls ($ (b2) + (0, 2) $) and ($ (d2) + (-2, 2) $) .. node [very near start] {$\epsilon / \epsilon$} (d2);
-    \draw (c2) .. controls ($ (c2) + (0, 1) $) and ($ (d2) + (-2, 1) $) .. node [very near start] {$\epsilon / \epsilon$} (d2);
+    \draw (b2) .. controls ($ (b2) + (0, 2) $) and ($ (d2) + (-2, 2) $) .. node [very near start] {$1 / \epsilon$} (d2);
+    \draw (c2) .. controls ($ (c2) + (0, 1) $) and ($ (d2) + (-2, 1) $) .. node [very near start] {$1 / \epsilon$} (d2);
     \path (b1) edge [draw=none] node [below=0.3in, midway] {
     $\begin{aligned}
     F \big( (0, 0, r^{1, m}) \big) \mid m \neq \infty &= (\Sigma, Q, x_1, \{y_m\}, \Delta) \\
@@ -234,25 +258,13 @@
             f_i &= F(r) = (\Sigma, Q_i, \{y_i\}, x_i, \Delta_i) \; \forall i = \overline{1, m} \\
             Q &= Q_1 \cup \hdots \cup Q_m \\
             \Delta &= \Delta_1 \cup \hdots \cup \Delta_m
-                \cup \big\{ (y_i, \epsilon, \epsilon, x_{i+1}), (y_i, \epsilon, \epsilon, y_m) \big\}_{i=1}^{m-1}
+                \cup \big\{ (y_i, 2, \epsilon, x_{i+1}), (y_i, 1, \epsilon, y_m) \big\}_{i=1}^{m-1}
     \end{aligned}$} (d2);
 \end{scope}
 
 \end{tikzpicture}
 
 
-\iffalse
-    \begin{align*}
-    \\
-    \\
-    F \big( (0, 0, r^{n, m}) \big) \mid n > 1 &= F \big( (0, 0, r \cdot r^{n - 1, m - 1})
-    \\
-    \\
-    \\
-    \\
-    \end{align*}
-\fi
-
 
 \end{document}
 

diff --git a/re2c/doc/tdfa_v2/part_1_tnfa.tex b/re2c/doc/tdfa_v2/part_1_tnfa.tex
index fddebbb51348995bd3232769fc2bab72a4c10311..dd4faccd3cddf42ef12f80c3841d29fa9709d4e3 100644 (file)
--- a/re2c/doc/tdfa_v2/part_1_tnfa.tex
+++ b/re2c/doc/tdfa_v2/part_1_tnfa.tex
@@ -781,45 +781,55 @@ dotted paths correspond to frames and the table entry is in bold.
     is a structure $(\Sigma, Q, F, q_0, \Delta)$, where:
     \begin{itemize}
         \item[] $\Sigma$ is a finite set of symbols (\emph{alphabet})
-%        \item[] $T$ is a finite set of \emph{tags}
-%        \item[] $P$ is a finite set of \emph{priorities}
         \item[] $Q$ is a finite set of \emph{states}
         \item[] $F \subseteq Q$ is the set of \emph{final} states
         \item[] $q_0 \in Q$ is the \emph{initial} state
 
+%        \item[] $T \subseteq \YN$ is a finite set of \emph{tags}
+%        \item[] $P \subseteq \YN$ is a finite set of \emph{priorities}
+
         \item[] $\Delta = \Delta^\Sigma \sqcup \Delta^\epsilon$ is the \emph{transition} relation, where:
         \begin{itemize}
-            \item[] $\Delta^\Sigma \subseteq Q \times \Sigma \times \{\epsilon\} \times Q$
-            \item[] $\Delta^\epsilon \subseteq Q \times \{\epsilon\} \times \big( (\YZ \times \YZ) \cup \{\epsilon\} \big) \times Q$
+            \item[] $\Delta^\Sigma \subseteq Q \times \Sigma \times \{\epsilon\} \times Q$ (transitions on symbols)
+            \item[] $\Delta^\epsilon \subseteq Q \times \YN \times \big( \YN \cup \{\epsilon\} \big) \times Q$ ($\epsilon$-transitions)
         \end{itemize}
-%        and all $\epsilon$-transitions from the same state have different priorities:
-%        $\forall (x, r, \epsilon, y), (\widetilde{x}, \widetilde{r}, \epsilon, \widetilde{y}) \in \Delta^\epsilon:
-%        x = \widetilde{x} \wedge y = \widetilde{y} \Rightarrow r \!\neq\! \widetilde{r}$.
+        and the property that all $\epsilon$-transitions from the same state have different priority:
+        $\forall (q, n, \Xund, \Xund), (q, m, \Xund, \Xund) \in \Delta^\epsilon: n \neq m$.
     \end{itemize}
     \end{Xdef}
 
 TNFA construction is given by the $F$ function defined on figure \ref{fig_tnfa}.
 
-    \begin{align*}
-    otag \big( (i, j, r) \big) &=
-        \begin{cases}
-            \epsilon         &\text{if } i = 0 \\[-0.5em]
-            (2i - 1, 2j - 1) &\text{if } i \neq 0
-        \end{cases}
-    \\
-    ctag \big( (i, j, r) \big) &=
-        \begin{cases}
-            \epsilon &\text{if } i = 0 \\[-0.5em]
-            (2i, 2j) \hphantom{{} - 1 {} - 1} &\text{if } i \neq 0
-        \end{cases}
-    \\
-    ntag \big( (i, j, r) \big) &=
-        \begin{cases}
-            \epsilon         &\text{if } i = 0 \\[-0.5em]
-            (1 - 2i, 1 - 2j) &\text{if } i \neq 0
-        \end{cases}
-    \end{align*}
+\begin{figure}\label{fig_tnfa}
+\includegraphics[width=\linewidth]{img/tnfa.pdf}
+\caption{
+TNFA construction.
+}
+\end{figure}
+
+\FloatBarrier
+
+%    \begin{align*}
+%    otag \big( (i, j, r) \big) =
+%        \begin{cases}
+%            \epsilon &\text{if } i = 0 \\[-0.5em]
+%            2i - 1   &\text{if } i \neq 0
+%        \end{cases}
+%    &&
+%    ctag \big( (i, j, r) \big) =
+%        \begin{cases}
+%            \epsilon &\text{if } i = 0 \\[-0.5em]
+%            2i       &\text{if } i \neq 0
+%        \end{cases}
+%    &&
+%    ntag \big( (i, j, r) \big) =
+%        \begin{cases}
+%            \epsilon  &\text{if } i = 0 \\[-0.5em]
+%            1 - 2i    &\text{if } i \neq 0
+%        \end{cases}
+%    \end{align*}
 
+\iffalse
 %    \begin{align*}
 %    F \big( (0, 0, r_1 \mid r_2) \big) &= (\Sigma, Q_1 \cup Q_2 \cup \{ x, y \}, x, \{y\}, \Delta) \\
 %        \text{where }
@@ -858,13 +868,7 @@ TNFA construction is given by the $F$ function defined on figure \ref{fig_tnfa}.
 %                \cup \{ (y_i, \epsilon, \epsilon, x_{i+1}), (y_i, \epsilon, \epsilon, y_m) \}_{i=1}^{m-1}
 %    \\
 %    \end{align*}
-
-\begin{figure}\label{fig_tnfa}
-\includegraphics[width=\linewidth]{img/tnfa.pdf}
-\caption{
-TNFA construction.
-}
-\end{figure}
+\fi
 
 
 
@@ -876,15 +880,15 @@ TNFA construction.
 
     \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
     \Fn {$\underline{match (\XN, \alpha_1 \dots \alpha_n)} \smallskip$} {
-        $(\Sigma, T, P, Q, F, q_0, T, \Delta) = \XN$ \;
+        $(\Sigma, Q, F, q_0, T, \Delta) = \XN$ \;
         $X = closure(\{ (\bot, q_0, \epsilon, t_0) \}, F, \Delta)$ \;
-        $(X, B, D) = step(0, X, B, D)$ \;
+        $(X, B, D) = step(0, X, P_1, P_2)$ \;
 
         \BlankLine
         \For {$i = \overline{1,n}$} {
             $X = reach(X, \Delta, \alpha_i)$ \;
-            $X = closure(X, B, D)$ \;
-            $(X, B, D) = step(i, X, B, D)$ \;
+            $X = closure(X, Q, F, \Delta, P_1, P_2)$ \;
+            $(X, P_1, P_2) = step(i, X, P_1, P_2)$ \;
         }
 
         \BlankLine
@@ -906,11 +910,11 @@ TNFA construction.
 
 
     \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
-    \Fn {$\underline{step(k, X, B, D)} \smallskip$} {
+    \Fn {$\underline{step(k, X, P_1, P_2)} \smallskip$} {
         let $\{ x_i \}_{i=1}^{n} = \{(q_i, p_i, u_i, t_i) \}_{i=1}^{n} = X$
 
         \BlankLine
-        \tcc {update tag values in $X$}
+%        \tcc {update tag values in $X$}
         \For {$i = \overline {1, n}$} {
             $b_1 \dots b_m = u_i$ \;
             \lFor {$j = \overline{1, m}$} {
@@ -919,17 +923,17 @@ TNFA construction.
         }
 
         \BlankLine
-        \tcc {update $B$ and $D$ matrices}
+%        \tcc {update $B$ and $D$ matrices}
         \For {$i = \overline {1, n}$} {
             \For {$j = \overline {1, n}$} {
-                $(h_1, h_2, l) = precedence (x_i, x_j, B, D)$ \;
-                $B' [p_i] [p_j] = h_1, \; D' [p_i] [p_j] = l$ \;
-                $B' [p_j] [p_i] = h_2, \; D' [p_j] [p_i] = -l$ \;
+                $(h_1, h_2, l) = precedence (x_i, x_j, P_1, P_2)$ \;
+                $P_1' [p_i] [p_j] = h_1, \; P_2' [p_i] [p_j] = l$ \;
+                $P_1' [p_j] [p_i] = h_2, \; P_2' [p_j] [p_i] = -l$ \;
             }
         }
 
         \BlankLine
-        \Return $(X, B', D')$ \;
+        \Return $(X, P_1', P_2')$ \;
     }
     \end{algorithm}