\Fn {$\underline{\ntag(r)} \smallskip$} {
$T = \{ t_1, \dots, t_n \} = T(r), n \geq 0$ \;
$Q = \{x_0, \dots, x_{2n}\}$ \;
- $\Delta = \{(x_{2i},1,1\!-\!2i, x_{2i+1}), (x_i, 1, -\!2i, x_{2i+2})\}_{i=0}^{n-1}$ \;
+ $\Delta = \{(x_{2i-2},1,1\!-\!2t_i, x_{2i-1}), (x_{2(n-i)-1}, 1, -\!2t_i, x_{2(n - i)})\}_{i=1}^{n}$ \;
\Return $(\Sigma, Q, T, \Delta, x_0, x_{2n})$ \;
}
\end{algorithm}
\end{figure*}
+
\begin{figure}\label{fig_tnfa_example}
\includegraphics[width=\linewidth]{img/tnfa_example.pdf}
\vspace{-2em}
\end{figure}
-%\begin{figure}\label{fig_tnfa}
-%\includegraphics[width=\linewidth]{img/tnfa.pdf}
-%\caption{
-%TNFA construction:
-%(a) -- operations of concatenation, union, closure on automata, \\
-%(b) -- subautomaton for submatch IRE,
-%(c) -- subautomaton for negative tags,
-%(d) -- (k) -- subautomata for non-submatch IRE, \\
-%(l) -- example TNFA for RE $(a|\epsilon)^{0,3}((a^{0,\infty})|(\epsilon))$.
-%}
-%\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 }
-% & F(r_1) = (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1), \;
-% F(r_2) = (\Sigma, Q_2, x_2, \{y_2\}, \Delta_2) \\
-% & \Delta = \Delta_1 \cup \Delta_2 \cup \big\{
-% (x, \epsilon, \epsilon, x_1), (y_1, \epsilon, ntag(r_2), y),
-% (x, \epsilon, ntag(r_1), x_2), (y_2, \epsilon, \epsilon, y)
-% \big\}
-% \\
-% F \big( (0, 0, r_1 \cdot r_2) \big) &= (\Sigma, Q_1 \cup Q_2 \setminus \{ x_2 \}, x_1, \{y_2\}, \Delta) \\
-% \text{where }
-% & F(r_1) = (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1), \;
-% F(r_2) = (\Sigma, Q_2, x_2, \{y_2\}, \Delta_2) \\
-% & \Delta = \Delta_1 \cup
-% \big\{ (y_1, \alpha, \beta, z) \mid (x_2, \alpha, \beta, z) \in \Delta_2 \big\}
-% \\
-% F \big( (0, 0, r^{n, m}) \big) \mid n > 1 &= F \big( (0, 0, r \cdot r^{n - 1, m - 1})
-% \\
-% F \big( (0, 0, r^{0, m}) \big) &= (\Sigma, Q_1 \cup \{ x, y \}, x, \{y\}, \Delta) \\
-% \text{where }
-% & F \big( (0, 0, r^{1, m}) \big) = (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1) \\
-% & \Delta = \Delta_1 \cup
-% \big\{ (x, \epsilon, \epsilon, x_1), (y_1, \epsilon, ntag(r_2), y), (x, \epsilon, ntag(r), y) \big\}
-% \\
-% F \big( (0, 0, r^{1, \infty}) \big) &= (\Sigma, Q_1 \cup \{ y \}, x_1, \{y\}, \Delta) \\
-% \text{where }
-% & F(r) = (\Sigma, Q_1, x_1, \{y_1\}, \Delta_1) \\
-% & \Delta = \Delta_1 \cup
-% \big\{ (y_1, \epsilon, \epsilon, x_1), (y_1, \epsilon, \epsilon, y) \big\}
-% \\
-% F \big( (0, 0, r^{1, m}) \big) \mid m \neq \infty &= (\Sigma, Q_1 \cup \hdots \cup Q_m, x_1, \{y_m\}, \Delta) \\
-% \text{where }
-% & f_i = F(r) = (\Sigma, Q_i, \{y_i\}, x_i, \Delta_i) \; \forall i = \overline{1, m} \\
-% & \Delta = \Delta_1 \cup \hdots \cup \Delta_m
-% \cup \{ (y_i, \epsilon, \epsilon, x_{i+1}), (y_i, \epsilon, \epsilon, y_m) \}_{i=1}^{m-1}
-% \\
-% \end{align*}
-\fi
-
-
-
\begin{figure*}
\begin{multicols}{2}
-
- \setstretch{0.85}
+ \setstretch{0.8}
+ \newcommand \ff {f\!\!f}
\begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
\Fn {$\underline{match (\XN, \alpha_1 \dots \alpha_n)} \smallskip$} {
}
\BlankLine
- \lIf {$\exists (\Xund, p, \Xund, t) \in X \mid p \in F$} {
- \Return $t$
+ \If {$\exists (\Xund, p, \Xund, t) \in X \mid p \in F$} {
+ \Return $output(t)$
} \lElse {
\Return $\bot$
}
\begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
\Fn {$\underline{reach (X, \alpha)} \smallskip$} {
- \Return $\{ (q, p, \epsilon, t) \mid (\Xund, q, \Xund, t) \in X \wedge
- (q, \alpha, \epsilon, p) \in \Delta^\Sigma \}$
+ \Return $\{ (q, p, \epsilon, t \cdot u) \mid $ \\
+ $\hspace{4em} (\Xund, q, u, t) \in X \wedge (q, \alpha, \epsilon, p) \in \Delta^\Sigma \}$
}
\end{algorithm}
\begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
\Fn {$\underline{step(k, X, P_1, P_2)} \smallskip$} {
- let $\{ x_i \}_{i=1}^{n} = \{(q_i, p_i, u_i, t_i) \}_{i=1}^{n} = X$
+ $\{ x_i \}_{i=1}^{n} = \{(q_i, p_i, u_i, t_i) \}_{i=1}^{n} = X$ \;
- \BlankLine
-% \tcc {update tag values in $X$}
- \For {$i = \overline {1, n}$} {
- $b_1 \dots b_m = u_i$ \;
- \For {$j = \overline{1, m}$} {
- \lIf {$b_j < 0$} {$t_i[b_j] = -1$}
- \lElse {$t_i[b_j] = k$}
- }
- }
-
- \BlankLine
-% \tcc {update $B$ and $D$ matrices}
\For {$i = \overline {1, n}$} {
\For {$j = \overline {1, n}$} {
$(h_1, h_2, l) = precedence (x_i, x_j, P_1, P_2)$ \;
$P_1' [p_i] [p_j] = h_1, \; P_2' [p_i] [p_j] = l$ \;
- $P_1' [p_j] [p_i] = h_2, \; P_2' [p_j] [p_i] = -l$ \;
+ $P_1' [p_j] [p_i] = h_2, \; P_2' [p_j] [p_i] = -l$
}
}
-
\BlankLine
\Return $(X, P_1', P_2')$ \;
}
\columnbreak
-
- \newcommand \ff {f\!\!f}
-
- \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
\Fn {$\underline {precedence (x, y, P_1, P_2)} \smallskip$} {
$(q, \Xund, \gamma, \Xund) = x$, where $\gamma = a_1 \dots a_n, \; n \geq 0$ \;
$(p, \Xund, \delta, \Xund) = y$, where $\delta = b_1 \dots b_m, \; m \geq 0$ \;
\Return $(hx, hy, l)$
}
\end{algorithm}
-
\end{multicols}
+\vspace{-2em}
\begin{center}
\caption{Matching algorithm.}
\end{center}
\end{figure*}
+\begin{figure*}
+\begin{multicols}{2}
+ \setstretch{0.8}
+ \newcommand \outputoffsets {out\!put \Xund o\!f\!\!f\!sets}
+ \newcommand \outputparsetree {out\!put \Xund parse \Xund tree}
+ \newcommand \parsetree {parse \Xund tree}
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
+ \Fn {$\underline{\outputparsetree(s)} \smallskip$} {
+ \Return $\parsetree(s,1)$
+ }
+ \end{algorithm}
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
+ \Fn {$\underline{\parsetree(s, i)} \smallskip$} {
+ \lIf {$s = \epsilon$} {
+ \Return $\epsilon^i$
+ }
+ \lIf {$s = \alpha \in \Sigma$} {
+ \Return $\alpha^i$
+ }
+ $a_0 \dots a_n = s$ where $n \geq 1$ \;
+ $j = k = 0$ \;
+ \While {$k < n$} {
+ $k' = k$ \;
+ $i' = (|a_{k'}| + 1) / 2$ \;
+ \lWhile {$|a_{k+1}| \geq 2i'$} {
+ $k = k + 1$
+ }
+ $j = j + 1$ \;
+ \lIf {$a_{k'} < 0$} {$t_j = \varnothing^{i'}$}
+ \lElse {$t_j = \parsetree (a_{k'+1} \dots a_{k-1}, i')$ }
+ }
+ \Return $T^i(t_1, \dots, t_j)$
+ }
+ \end{algorithm}
+
+ \columnbreak
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
+ \Fn {$\underline{\outputoffsets(s)} \smallskip$} {
+ $a_1 \dots a_n = s$ \;
+ $pos = 0$ \;
+ \For {$i = \overline {1, n}$} {
+ \lIf {$a_i \in \Sigma$} {
+ $pos = pos + 1$
+ }
+ \Else {
+ $p = pos$ \;
+ \lIf {$a_i < 0$} {$p = -1$}
+ \lIf {$a_i mod 2 \equiv 0$} {$pmatch[|a_i|/2].rm \Xund so = p$}
+ \lElse {$pmatch[(|a_i| - 1)/2].rm \Xund eo = p$}
+ }
+ }
+ \Return $pmatch$ \;
+ }
+ \end{algorithm}
+\end{multicols}
+\vspace{-2em}
+\begin{center}
+\caption{Two possible $output()$ functions.}
+\end{center}
+\end{figure*}
+
+
+
\clearpage
\pagebreak
projects / re2c / commitdiff