Paper: tweaked TNFA construction.

diff --git a/re2c/doc/tdfa_v2/img/tnfa_construction.tex b/re2c/doc/tdfa_v2/img/tnfa_construction.tex
index e900bc94e03eadae1415d3d34b69f35c01e3e9f6..b4223c7d1f78ac7208f2a20affd79c1c8ea9262a 100644 (file)
--- a/re2c/doc/tdfa_v2/img/tnfa_construction.tex
+++ b/re2c/doc/tdfa_v2/img/tnfa_construction.tex
@@ -36,7 +36,7 @@
 \begin{scope}[xshift=0in, yshift=0in]
 
 \begin{scope}[xshift=0in, yshift=0in]
-    \node[draw=none] (a) {$\hphantom{\hspace{2in}}$};
+    \node[draw=none] (a) {$\hphantom{\hspace{1in}}$};
 \end{scope}
 
 \begin{scope}[xshift=0in, yshift=0in]
@@ -55,34 +55,34 @@
     $\retonfa \big( (0, 0, a), y \big) $}] (a) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-1.5in]
+\begin{scope}[xshift=0in, yshift=-1.4in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
     \node[state] (a1) {$z$};
-    \node[style1, right of = a1] (a) {$F(r_1)$};
+    \node[style1, right of = a1] (a) {$F(s)$};
     \node[style2, right of = a] (a2) {$x$};
-    \node[style1, right of = a2] (b) {$F(r_2)$};
+    \node[style1, right of = a2] (b) {$F(u)$};
     \node[style2, right of = b] (b2) {$y$};
     \node [label={[label distance=0.2in, below left]270:\large{(c)}}] (a) {};
     \node [label={[label distance=0.2in, below right]270:
-    $\retonfa \big( (0, 0, r_1 \cdot r_2), y \big) $}] (a1) {};
+    $\retonfa \big( (0, 0, s \cdot u), y \big) $}] (a1) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-2.8in]
+\begin{scope}[xshift=0in, yshift=-2.6in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
     \node[state] (a) {$x$};
     \node[state, above right of = a, yshift = -0.35in] (b1) {$x_2$};
-    \node[style1, right of = b1] (b) {$F(r_1)$};
+    \node[style1, right of = b1] (b) {$F(s)$};
     \node[style2, right of = b] (b2) {$x_1$};
-    \node[style1, right of = b2, rotate around={-21:(b2)}] (d) {$N(r_2)$};
+    \node[style1, right of = b2, rotate around={-21:(b2)}] (d) {$N(u)$};
 
     \node[state, below right of=a, yshift = 0.35in] (c1) {$x_4$};
-    \node[style1, right of = c1] (c2) {$N(r_1)$};
+    \node[style1, right of = c1] (c2) {$N(s)$};
     \node[style2, right of = c2] (c3) {$x_3$};
-    \node[style1, right of = c3, rotate around={21:(c3)}] (c) {$F(r_2)$};
+    \node[style1, right of = c3, rotate around={21:(c3)}] (c) {$F(u)$};
     \node[style2, right of = c, rotate around={21:(c)}] (d) {$y$};
     \path
         (a)  edge [bend left]  node {$1 / \epsilon$} (b1)
@@ -90,10 +90,10 @@
     ;
     \node [label={[label distance=0.5in, below left]270:\large{(d)}}] (a) {};
     \node [label={[label distance=0.5in, below right]270:
-    $\retonfa \big( (0, 0, r_1 \mid r_2), y \big) $}] (a) {};
+    $\retonfa \big( (0, 0, s \mid u), y \big) $}] (a) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-4in]
+\begin{scope}[xshift=0in, yshift=-3.8in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
@@ -107,7 +107,7 @@
     $\retonfa \big( (0, 0, r^{n, m}), y \big) \mid_{n \;>\; 1} $}] (a1) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-5.2in]
+\begin{scope}[xshift=0in, yshift=-5in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
@@ -127,7 +127,7 @@
     $\retonfa \big( (0, 0, r^{0, m}), y \big) $}] (a) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-6.8in]
+\begin{scope}[xshift=0in, yshift=-6.6in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
@@ -146,7 +146,7 @@
     $\retonfa \big( (0, 0, r^{1, \infty}), y \big) $}] (b1) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-7.75in]
+\begin{scope}[xshift=0in, yshift=-7.4in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
@@ -159,7 +159,7 @@
     $\retonfa \big( (0, 0, r^{1, 1}), y \big) $}] (b1) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-9in]
+\begin{scope}[xshift=0in, yshift=-8.6in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
@@ -180,7 +180,7 @@
     $\retonfa \big( (0, 0, r^{1, m}), y \big) \mid_{1 < m < \infty} $}] (b1) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-10.5in]
+\begin{scope}[xshift=0in, yshift=-10in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
@@ -198,28 +198,27 @@
     $\retonfa \big( (i, \Xund, r) \big) \mid_{i \;\neq\; 0} $}] (a) {};
 \end{scope}
 
-\begin{scope}[xshift=0in, yshift=-11.2in]
+\begin{scope}[xshift=0in, yshift=-10.8in]
     \def\offs{-0.5in}
     \def\widd{1.3in}
 
-    \node[state, accepting] (a) {$y$};
-    \node [label={[label distance=0.1in, below left]270:\large{(j)}}] (a) {};
-    \node [label={[label distance=0.1in, below right]270:
-    $\ntag \big( (0, \Xund, \Xund) \big) $}] (a) {};
-\end{scope}
-
-\begin{scope}[xshift=0in, yshift=-12in]
-    \def\offs{-0.5in}
-    \def\widd{1.3in}
-
-    \node[state] (a) {$x$};
-    \node[state, accepting, right of = a] (a1) {$y$};
+    \node[state] (a) {$x_0$};
+    \node[state, right of = a] (a1) {$x_1$};
+    \node[state, right of = a1] (b1) {$x_2$};
+    \node[draw=none, right of = b1, xshift=-0.5in] (b) {\large{$\dots$}};
+    \node[state, accepting, right of = b, xshift=-0.5in] (b2) {$x_{2n}$};
     \path
-        (a) edge node {$1 / 1 -\! 2i $} (a1)
+        (a) edge node {$1 / -\!(2i-1) $} (a1)
+        (a1) edge node {$1 / -\! 2i $} (b1)
+        (b1) edge node {} (b)
+        (b) edge node {} (b2)
     ;
-    \node [label={[label distance=0.1in, below left]270:\large{(k)}}] (a) {};
+    \node [label={[label distance=0.1in, below left]270:\large{(j)}}] (a) {};
     \node [label={[label distance=0.1in, below right]270:
-    $\ntag \big( (i, \Xund, r) \big) \mid_{i \;\neq\; 0} $}] (a) {};
+    $\begin{aligned}
+        N \big( (i, \Xund, r) \big)
+    \end{aligned}$
+    }] (a) {};
 \end{scope}
 
 \end{scope}

diff --git a/re2c/doc/tdfa_v2/img/tnfa_example.tex b/re2c/doc/tdfa_v2/img/tnfa_example.tex
index 08bbed66fcfb5f2df2b542dd55bebed117a063a8..db243066f4fb2c75abd14ff2f43737716ea15bd0 100644 (file)
--- a/re2c/doc/tdfa_v2/img/tnfa_example.tex
+++ b/re2c/doc/tdfa_v2/img/tnfa_example.tex
@@ -20,9 +20,9 @@
 
 \tikzstyle{every node}=[]
 \tikzstyle{every state}=[circle
-    , minimum size=0.15in
+    , minimum size=0.16in
     , rectangle
-    , rounded corners=4
+    , rounded corners=5
     , inner sep = 2pt
     , outer sep = 0pt
     , node distance = 0.4in]
@@ -30,8 +30,8 @@
 \newcommand{\zz}{0.06in}
 
 \begin{scope}[xshift=-0.8in, yshift=-10.5in]
-    \footnotesize
-    %\scriptsize
+    %\footnotesize
+    \scriptsize
 
     % ((epsilon|a*)((a|epsilon){0,3}))
 
@@ -47,6 +47,7 @@
     \node[state, below right of = x21] (x22) {$8$};
 
     \node[state, right of = x1, xshift=0.6in] (z1) {$18$};
+    \node[state, right of = z1, xshift=0.27in] (z2) {$19$};
 
     \node[state, right of = x22] (y15) {$9$};
     \node[state, above right of = y15] (y16) {$10$};
@@ -58,27 +59,29 @@
     \node[state, above right of = z15] (z16) {$15$};
     \node[state, fill=lightgray, above right of = z16, xshift = \zz, yshift = -\zz] (z17) {$16$};
     \node[state, below right of = z17, xshift = \zz, yshift = +\zz] (z21) {$17$};
-    \node[state, below right of = z21] (z22) {$19$};
+    \node[state, below right of = z21] (z22) {$20$};
 
-    \node[state, below right of = z22] (x23) {$20$};
+    \node[state, below right of = z22] (x23) {$21$};
 
-    \node[state, above right of = x23] (x2) {$21$};
-    \node[state, above right of = x2, xshift = \zz, yshift = -\zz] (x4X) {$22$};
-    \node[state, above right of = x4X] (x4) {$23$};
+    \node[state, above right of = x23] (x2) {$22$};
+    \node[state, above right of = x2, xshift = \zz, yshift = -\zz] (x4X) {$23$};
+    \node[state, above right of = x4X] (x4) {$24$};
 
-    \node[state, below right of = x2, xshift = \zz+0.1in, yshift = +\zz] (x3) {$28$};
-    \node[state, right of = x3, xshift=0.1in] (x3Z) {$29$};
-    \node[state, above right of = x3Z] (x3X) {$30$};
+    \node[state, below right of = x2, xshift = \zz+0.1in, yshift = +\zz] (x3) {$30$};
+    \node[state, right of = x3, xshift=0.1in] (x3Y) {$31$};
+    \node[state, right of = x3Y, xshift=0.1in] (x3Z) {$32$};
+    \node[state, above right of = x3Z] (x3X) {$33$};
     \node[state, below right of = x3X, draw=none, inner sep=0, minimum size=0] (x3W) {};
 
-    \node[state, fill=lightgray, above right of = x4, xshift = \zz, yshift = -\zz] (x5) {$24$};
-    \node[state, right of = x5] (x6) {$25$};
-    \node[state, below right of = x6, xshift = \zz, yshift = \zz] (x7) {$26$};
-    \node[state, below right of = x7] (x7Y) {$27$};
-    \node[state, below right of = x7Y, xshift = \zz, yshift = +\zz] (x12) {$31$};
-    \node[state, below right of = x12] (x14) {$32$};
+    \node[state, fill=lightgray, above right of = x4, xshift = \zz, yshift = -\zz] (x5) {$25$};
+    \node[state, right of = x5] (x6) {$26$};
+    \node[state, below right of = x6, xshift = \zz, yshift = \zz] (x7) {$27$};
+    \node[state, below right of = x7] (x7X) {$28$};
+    \node[state, right of = x7X] (x7Y) {$29$};
+    \node[state, below right of = x7Y, xshift = \zz, yshift = +\zz] (x12) {$34$};
+    \node[state, below right of = x12] (x14) {$35$};
 
-    \node[state, fill=lightgray, accepting, below right of = x14] (x24) {$33$};
+    \node[state, fill=lightgray, accepting, below right of = x14] (x24) {$36$};
 
     \path
         (x0) edge node [above left] {$1/1$} (x1)
@@ -86,6 +89,7 @@
         (z0) edge node {$1/\epsilon$} (x15)
 
         (z0) edge [bend right=20] node [above, near end] {$2/\epsilon$} (z1)
+        (z1) edge node [below] {$1/\!\!-\!\!5$} (z2)
 
         (x15) edge node [above left] {$1/5$} (x16)
         (x16) edge [bend left]  node [above] {$1/\epsilon$} (x17)
@@ -114,15 +118,17 @@
         (x23) edge node [below right] {$1/7$} (x2)
         (x2) edge [bend left] node [above] {$1/\epsilon$} (x4X)
         (x2) edge [bend right] node [above] {$\quad 2/\epsilon$} (x3)
-        (x3) edge node [above] {$2/\!\!-\!\!9$} (x3Z)
-        (x3Z) edge node [below right] {$1/11$} (x3X)
+        (x3) edge node [below] {$2/\!\!-\!\!9$} (x3Y)
+        (x3Y) edge node [below] {$2/\!\!-\!\!10$} (x3Z)
+        (x3Z) edge node [above left] {$1/11$} (x3X)
         (x4X) edge node {$1/9$} (x4)
         (x4) edge [bend left] node [above] {$1/\epsilon$} (x5)
         (x4) edge [bend right=20] node [above] {$2/\epsilon$} (x7)
         (x5) edge node [above] {$a/\epsilon$} (x6)
         (x6) edge [bend left] node [above] {$2/\epsilon$} (x7)
-        (x7) edge node {$1/10$} (x7Y)
-        (x7Y) edge [bend left] node {$1/\!\!-\!\!11$} (x12)
+        (x7) edge node {$1/10$} (x7X)
+        (x7X) edge node {$1/\!\!-\!\!11$} (x7Y)
+        (x7Y) edge [bend left] node {$1/\!\!-\!\!12$} (x12)
         (x12) edge node [above right] {$1/8$} (x14)
 
         (x14) edge node [above right] {$1/2$} (x24)
@@ -130,8 +136,10 @@
 
     \draw (y22) .. controls ($ (y22) + (0.5, -0.6) $) and ($ (z22) + (-0.5, -0.6) $) .. node [above] {$1/\epsilon$} (z22);
     \draw (x22) .. controls ($ (x22) + (0.8, -0.8) $) and ($ (z22) + (-0.8, -0.8) $) .. node [above, near start] {$1/\epsilon$} (z22);
-    \draw (z1) .. controls ($ (z1) + (2, 0) $) and ($ (z22) + (-1.0, -1.0) $) .. node [above, very near start] {$1/\!\!-\!\!5$} (z22);
+    \draw (z2) .. controls ($ (z2) + (2, 0) $) and ($ (z22) + (-1.0, -1.0) $) .. node [below, very near start] {$1/\!\!-\!\!6$} (z22);
     \draw (x6) .. controls ($ (x6) + (0.3, 0.8) $) and ($ (x5) + (-0.3, 0.8) $) .. node [above] {$1/\epsilon$} (x5);
+
+%    \draw (x3X) .. controls ($ (x3X) + (0.9, -0.9) $) and ($ (x3X) + (0.5, -0.9) $) .. node [above right, near start] {$1/12$} (x12);
     \path[-] (x3X) edge node {$1/12$} (x3W);
     \path (x3W) edge [bend right=20] node {} (x12);
 

diff --git a/re2c/doc/tdfa_v2/part_1_tnfa.tex b/re2c/doc/tdfa_v2/part_1_tnfa.tex
index e097523a721574ec4df0724bd473c08737f2bad7..e8cf8d3767cc08c804b8d87543e3f0467ff8e247 100644 (file)
--- a/re2c/doc/tdfa_v2/part_1_tnfa.tex
+++ b/re2c/doc/tdfa_v2/part_1_tnfa.tex
@@ -781,7 +781,7 @@ TNFA construction is given by the $F$ function defined on figure \ref{fig_tnfa}.
 \begin{figure*}
 \begin{multicols}{2}
 \fontsize{8}{10}
-    \setstretch{0.85}
+    \setstretch{0.8}
 
     %\fontsize{9.5pt}{11.5pt}\selectfont
 
@@ -790,97 +790,100 @@ TNFA construction is given by the $F$ function defined on figure \ref{fig_tnfa}.
     \newcommand \retonfa {F}
     \newcommand \ntag {N}
 
-    \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+    \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
     \Fn {$\underline{\retonfa(r, y)} \smallskip$} {
         \If {$r = (0, 0, \epsilon)$} {
-            \Return $(\Sigma, \{y\}, \emptyset, \emptyset, y, y)$ \;
+            \Return $(\Sigma, \{y\}, \emptyset, \emptyset, y, y)$
         }
         \BlankLine
         \ElseIf {$r = (0, 0, \alpha) \mid_{\alpha \in \Sigma}$} {
-            \Return $(\Sigma, \{x,y\}, \emptyset, \{(x, \alpha, \epsilon, y)\}, x, y)$ \;
+            \Return $(\Sigma, \{x,y\}, \emptyset, \{(x, \alpha, \epsilon, y)\}, x, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1 \cdot r_2)$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, x, y) = \retonfa (r_1, y)$ \;
-            $(\Sigma, Q_2, T_2, \Delta_2, z, x) = \retonfa (r_2, x)$ \;
-            \Return $(\Sigma, Q_1 \cup Q_2, T_1 \cup T_2, \Delta_1 \cup \Delta_2, z, y)$ \;
+        \ElseIf {$r = (0, 0, s \cdot u)$} {
+            $(\Sigma, Q_s, T_s, \Delta_1, x, y) = \retonfa (s, y)$ \;
+            $(\Sigma, Q_u, T_u, \Delta_2, z, x) = \retonfa (u, x)$ \;
+            \Return $(\Sigma, Q_s \cup Q_u, T_s \cup T_u, \Delta_s \cup \Delta_u, z, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1 \mid r_2)$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, x_1, y) = \ntag (r_2, y)$ \;
-            $(\Sigma, Q_2, T_2, \Delta_2, x_2, x_1) = \retonfa (r_1, x_1)$ \;
-            $(\Sigma, Q_3, T_3, \Delta_3, x_3, y) = \retonfa (r_2, y)$ \;
-            $(\Sigma, Q_4, T_4, \Delta_4, x_4, x_3) = \ntag (r_1, x_3)$ \;
+        \ElseIf {$r = (0, 0, s \mid u)$} {
+            $(\Sigma, Q_1, T_1, \Delta_1, x_1, y) = \ntag (u, y)$ \;
+            $(\Sigma, Q_2, T_2, \Delta_2, x_2, x_1) = \retonfa (s, x_1)$ \;
+            $(\Sigma, Q_3, T_3, \Delta_3, x_3, y) = \retonfa (u, y)$ \;
+            $(\Sigma, Q_4, T_4, \Delta_4, x_4, x_3) = \ntag (s, x_3)$ \;
             $Q = Q_1 \cup Q_2 \cup Q_3 \cup Q_4 \cup \{x\}$ \;
             $T = T_1 \cup T_2 \cup T_3 \cup T_4$ \;
             $\Delta = \Delta_1 \cup \Delta_2 \cup \Delta_3 \cup \Delta_4 \cup \{(x,1,\epsilon,x_2), (x,2,\epsilon,x_4)\}$ \;
-            \Return $(\Sigma, Q, T, \Delta, x, y)$ \;
+            \Return $(\Sigma, Q, T, \Delta, x, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1^{n,m}) \mid_{n > 0}$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, x, y) = \retonfa ((0, 0, r_1^{n-1,m}), y)$ \;
-            $(\Sigma, Q_2, T_2, \Delta_2, z, x) = \retonfa (r_1, x)$ \;
-            \Return $(\Sigma, Q_1 \cup Q_2, T_1 \cup T_2, \Delta_1 \cup \Delta_2, z, y)$ \;
+        \ElseIf {$r = (0, 0, s^{n,m}) \mid_{n > 0}$} {
+            $(\Sigma, Q_1, T_1, \Delta_1, x, y) = \retonfa ((0, 0, s^{n-1,m}), y)$ \;
+            $(\Sigma, Q_2, T_2, \Delta_2, z, x) = \retonfa (s, x)$ \;
+            \Return $(\Sigma, Q_1 \cup Q_2, T_1 \cup T_2, \Delta_1 \cup \Delta_2, z, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1^{0,m})$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, x_1, y) = \retonfa ((0, 0, r_1^{1,m}), y)$ \;
-            $(\Sigma, Q_2, T_2, \Delta_2, x_2, y) = \ntag (r_1, y)$ \;
+        \ElseIf {$r = (0, 0, s^{0,m})$} {
+            $(\Sigma, Q_1, T_1, \Delta_1, x_1, y) = \retonfa ((0, 0, s^{1,m}), y)$ \;
+            $(\Sigma, Q_2, T_2, \Delta_2, x_2, y) = \ntag (s, y)$ \;
             $Q = Q_1 \cup Q_2 \cup \{x\}$ \;
             $\Delta = \Delta_1 \cup \Delta_2 \cup \{(x,1,\epsilon,x_1), (x,2,\epsilon,x_2)\}$ \;
-            \Return $(\Sigma, Q, T_1 \cup T_2, \Delta, x, y)$ \;
+            \Return $(\Sigma, Q, T_1 \cup T_2, \Delta, x, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1^{1,\infty})$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, z, x) = \retonfa (r_1, \Xund)$ \;
+        \ElseIf {$r = (0, 0, s^{1,\infty})$} {
+            $(\Sigma, Q_1, T_1, \Delta_1, z, x) = \retonfa (s, \Xund)$ \;
             $Q = Q_1 \cup \{y\}$ \;
             $\Delta = \Delta_1 \cup \{(x,1,\epsilon,z), (x,2,\epsilon,y)\}$ \;
-            \Return $(\Sigma, Q, T_1, \Delta, z, y)$ \;
+            \Return $(\Sigma, Q, T_1, \Delta, z, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1^{1,1})$} {
-            \Return $\retonfa (r_1, y)$ \;
+        \ElseIf {$r = (0, 0, s^{1,1})$} {
+            \Return $\retonfa (s, y)$
         }
         \BlankLine
-        \ElseIf {$r = (0, 0, r_1^{1,m}) \mid_{1 < m < \infty}$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, x, y) = \retonfa ((0, 0, r_1^{1,m-1}), y)$ \;
+        \ElseIf {$r = (0, 0, s^{1,m}) \mid_{1 < m < \infty}$} {
+            $(\Sigma, Q_1, T_1, \Delta_1, x, y) = \retonfa ((0, 0, s^{1,m-1}), y)$ \;
             $(\Sigma, Q_2, T_2, \Delta_2, w, z) = \retonfa (r_1, \Xund)$ \;
             $\Delta = \Delta_1 \cup \Delta_2 \cup \{(z,1,\epsilon,y), (z,2,\epsilon,x)\}$ \;
-            \Return $(\Sigma, Q_1 \cup Q_2, T_1 \cup T_2, \Delta, w, y)$ \;
+            \Return $(\Sigma, Q_1 \cup Q_2, T_1 \cup T_2, \Delta, w, y)$
         }
         \BlankLine
-%        \ElseIf {$r = (0, 0, r_1^{1,m}) \mid_{1 < m < \infty}$} {
-%            $y_m = y$ \;
-%            \For {$i = \overline{1,m}$} {
-%                $(\Sigma, Q_i, T_i, \Delta_i, x_i, y_i) = \retonfa (r_1, y_i)$ \;
-%            }
-%            $Q = \bigcup\nolimits_{i=1}^m Q_i$
-%            $\Delta = \bigcup\nolimits_{i=1}^m \Delta_i \cup \{(y_i,1,\epsilon,y_{i+1}),(y_i,2,\epsilon,x_{i+1})\}_{i=1}^m$
-%            \Return $(\Sigma, Q, T, \Delta, x_1, y)$ \;
-%        }
-        \BlankLine
-        \ElseIf {$r = (i, \Xund, r_1) \mid_{i \neq 0}$} {
-            $(\Sigma, Q_1, T_1, \Delta_1, z, x) = \retonfa (r_1, \Xund)$ \;
+        \ElseIf {$r = (i, \Xund, s) \mid_{i \neq 0}$} {
+            $(\Sigma, Q_1, T_1, \Delta_1, z, x) = \retonfa (s, \Xund)$ \;
             $Q = Q_1 \cup \{w, y\}$ \;
             $T = T_1 \cup \{2i\!-\!1, 2i\}$ \;
             $\Delta = \Delta_1 \cup \{(w,1,2i\!-\!1,z), (x,1,2i,y)\}$ \;
-            \Return $(\Sigma, Q, T, \Delta, w, y)$ \;
+            \Return $(\Sigma, Q, T, \Delta, w, y)$
         }
-
     }
     \end{algorithm}
 
 
-    \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+    \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
     \Fn {$\underline{\ntag(r)} \smallskip$} {
-        \If {$r = (0, 0, \Xund)$} {
-            \Return $(\Sigma, \{y\}, \emptyset, \emptyset, y, y)$ \;
+        $T = \{ t_1, \dots, t_n \} = T(r), n \geq 0$ \;
+        $Q = \{x_0, \dots, x_{2n}\}$ \;
+        $\Delta = \{(x_{2i},1,1\!-\!2i, x_{2i+1}), (x_i, 1, -\!2i, x_{2i+2})\}_{i=0}^{n-1}$ \;
+        \Return $(\Sigma, Q, T, \Delta, x_0, x_{2n})$ \;
+    }
+    \end{algorithm}
+
+
+    \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
+    \Fn {$\underline{T((i, \Xund, r))} \smallskip$} {
+        \lIf {$i = 0$} {
+            \Return $\emptyset$
         }
-        \BlankLine
-        \ElseIf {$r = (i, \Xund, \Xund) \mid_{i \neq 0}$} {
-            \Return $(\Sigma, \{x,y\}, \{1\!-\!2i\}, \{(x, 1, 1\!-\!2i, y)\}, x, y)$ \;
+        \lElseIf {$r = s \mid u \vee r = s \cdot u$} {
+            \Return $\{i\} \cup T(s) \cup T(u)$
+        }
+        \lElseIf {$r = s^{n,m}$} {
+            \Return $\{i\} \cup T(s)$
+        }
+        \lElse {
+            \Return $\{i\}$
         }
-
     }
     \end{algorithm}
 
@@ -892,17 +895,14 @@ TNFA construction is given by the $F$ function defined on figure \ref{fig_tnfa}.
 
 
 \end{multicols}
-%\vspace{-2em}
-%\includegraphics[width=\linewidth]{img/tnfa_example.pdf}
 \vspace{-2em}
-\begin{center}
 \caption{TNFA construction.}
-\end{center}
 \end{figure*}
 
 
 \begin{figure}\label{fig_tnfa_example}
 \includegraphics[width=\linewidth]{img/tnfa_example.pdf}
+\vspace{-2em}
 \caption{
     Example TNFA for RE $(a|\epsilon)^{0,3}((a^{0,\infty})|(\epsilon))$.
 }