
diff -r 660cf698eb26 -r 3831621d7b14 ChengsongTanPhdThesis/Chapters/Bitcoded1.tex
--- a/ChengsongTanPhdThesis/Chapters/Bitcoded1.tex	Tue Jul 25 17:28:29 2023 +0100
+++ b/ChengsongTanPhdThesis/Chapters/Bitcoded1.tex	Wed Aug 23 03:02:31 2023 +0100
@@ -44,288 +44,351 @@
 \end{tabular}
 \end{center}
 \noindent
-The first problem with this algorithm is that
-for the function $\inj$ to work properly
-we cannot destroy the structure of a regular expression,
-and therefore cannot simplify too aggressively.
-For example,
-\[
-	r + (r + a) \rightarrow r + a
-\]
-cannot be applied because that would require
-breaking up the inner alternative.
-The $\lexer$ plus $\textit{simp}$ therefore only enables
-same-level de-duplications like
-\[
-	r + r \rightarrow r.
-\]
-Secondly, the algorithm recursively calls $\lexer$ on 
-each new character input,
-and before starting a child call
-it stores information of previous lexing steps
-on a stack, in the form of regular expressions
-and characters: $r_0$, $c_0$, $r_1$, $c_1$, etc.
-Each descent into deeper recursive calls in $\lexer$
-causes a new pair of $r_i, c_i$ to be pushed to the call stack.
-\begin{figure}[H]
-\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
-%\draw (-6,-6) grid (6,6);
-\node  [ circle ] (r) at (-6, 5) {$r$};
+This algorithm works nicely as a functional program that utilizes Brzozowski derivatives:
+each derivative character is remembered and stacked up, and
+injected back in reverse order as they have been taken derivative of.
+The derivative operation $\backslash$ and its reverse operation
+$\inj$ is of similar shape and compexity, and work in lockstep with each other.
+However if we take a closer look into the example run 
+of $\lexer$ we have shown in chapter \ref{Inj},
+many inefficiencies exist:
+\begin{center}
+	\begin{tabular}{lcl}
+		$(a+\textcolor{magenta}{a}\textcolor{blue}{a})^* \cdot \textcolor{red}{c}$ & $\stackrel{\backslash \textcolor{magenta}{a}}{\rightarrow}$ & $((\ONE + \textcolor{magenta}{\ONE} \textcolor{blue}{a})\cdot (a+aa)^*)\cdot \textcolor{red}{c}+\ZERO$\\
+				   %& $\stackrel{\rightarrow}{\backslash a}$ & $((\ONE + \ONE a)\cdot (a+aa)^*)\cdot c + \ZERO$\\
+				    & $\stackrel{\backslash \textcolor{blue}{a}}{\rightarrow}$  &
+				    $(((\ZERO+(\ZERO a+ \textcolor{blue}{\ONE}))\cdot (a+aa)^* + (\ONE+\ONE a)\cdot (a+aa)^* )\cdot \textcolor{red}{\mathbf{c}} + \ZERO)+\ZERO$\\
+				    & $\stackrel{\backslash \textcolor{red}{c}}{\rightarrow}$  &
+				    $((r_{=0}\cdot c + \textcolor{red}{\ONE})+\ZERO)+\ZERO$\\
+				    & $\stackrel{\mkeps}{\rightarrow}$ & $\Left (\Left \; (\Right \; \textcolor{red}{\Empty}))$ \\
+				   & $\stackrel{\inj \;\textcolor{red}{c} }{\rightarrow}$ & 
+				   $\Left \; (\Left \; (\Seq \;(\Left \; (\Seq \; (\Right \; (\Right\; \textcolor{blue}{\Empty})) \; \Stars \, [])) \; \textcolor{red}{c}))$\\
+				   & $\stackrel{\inj \;\textcolor{blue}{a}}{\rightarrow}$ & 
+				   $\Left\; (\Seq \; (\Seq\; (\Right \; (\Seq \; \textcolor{magenta}{\Empty }\; \textcolor{blue}{ \mathbf{a}}) )
+				   \;\Stars \,[]) \; \textcolor{red}{c})$\\
+				   & $\stackrel{\inj \;\textcolor{magenta}{a}}{\rightarrow}$ & $\Seq \; (\Stars \; [\Right \; (\Seq \; \mathbf{\textcolor{magenta}{a}} \; \textcolor{blue}{a})]) \; \textcolor{red}{ c}$
 
-%\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
-\node  [circle, minimum size = 0.1, draw] (c1) at (-4, 5.4) {$c_1$};
-%
-%\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
-\node  [minimum size = 0.5, circle, draw] (r1) at (-2, 5) {$r_1$};
-
-\node [minimum width = 2cm, rectangle, draw] (stack) at (0, 3) {Stack};
+				   %$\inj \; r \; c\A$
+		%$\inj \; (a\cdot b)\cdot c \; \Seq \; \ONE \; b$ & $=$ & $(a+e)$\\
+	\end{tabular}
+\end{center}
+\noindent
+For the un
 
-\path
-	(r)
-        edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
-
-\path   (r1)
-	edge [bend right, dashed] node {saved} (stack);
-\path   (c1)
-	edge [bend right, dashed] node {} (stack);
-
-
-\end{tikzpicture}
-\caption{First derivative taken}
-\end{figure}
+\begin{center}
+	\begin{tabular}{lcl}
+$\inj$ &  $\quad ((\ONE + {\ONE} 
+	\textcolor{blue}{a})\cdot (a+aa)^*)\cdot
+	 + \ZERO \; \quad \textcolor{blue}{a} \; $ & \\
+$\quad \Left \; 
+(\Left \; (\Seq \;(\Left \; (\Seq \; (\Right \; (\Right\; 
+	\textcolor{blue}{\Empty})) \; \Stars \, [])) \; c))$ & \\$=$\\
+	$\Left \; (\inj \; ((\ONE + \ONE 
+	\textcolor{blue}{a})\cdot (a+aa)^*)\cdot 
+	c \quad  \textcolor{blue}{a} \quad (\Left \; (\Seq\; ( \Left \; (\Seq \; (\Right \; (\Right\; 
+	\textcolor{blue}{\Empty})) \; \Stars \, []) ) \; c ) )\;\;)$ &\\ $=$\\
+	$\Left \; (\Seq \; (\inj \; (\ONE + \ONE \textcolor{blue}{a})\cdot(a+aa)^* \quad \textcolor{blue}{a} \quad
+	(\Left \; (\Seq \; (\Right \; (\Right\;\textcolor{blue}{\Empty})))) ) \; c ) $ & \\$=$\\
+	$\Left \; (\Seq \; (\Seq \; (\inj \quad (\ONE + \ONE \textcolor{blue}{a}) \quad \textcolor{blue}{a}\quad \Right \;(\Right \; \textcolor{blue}{\Empty}) ) \; \Stars \,[])\; c)$ &\\ $=$ \\
+	$\Left \; (\Seq \; (\Seq \; (\Right \; (\inj \; \ONE \textcolor{blue}{a} \quad \textcolor{blue}{a}\quad (\Right \;\textcolor{blue}{\Empty})))) \; \Stars \, [])$
+																		   & \\ $=$\\
+																		   $\Left \; (\Seq \; (\Seq \; (\Right \; (\Seq \; (\mkeps \; \ONE)\;(\inj \;\textcolor{blue}{a} \; \textcolor{blue}{a} \; \textcolor{blue}{\Empty})) ) ) \; \Stars \, [] )$
+																		   & \\ $=$\\
+																		   $\Left \; (\Seq \; (\Seq \; (\Right \; (\Seq \; \Empty \; \textcolor{blue}{a}) ) ) \; \Stars \, [] )$
+ 	\end{tabular}
+\end{center}
 
 
 
-\begin{figure}[H]
-\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
-%\draw (-6,-6) grid (6,6);
-\node  [ circle ] (r) at (-6, 5) {$r$};
 
-%\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
-\node  [circle, minimum size = 0.1, ] (c1) at (-4, 5.4) {$c_1$};
-%
-%\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
-\node  [minimum size = 0.5, circle, ] (r1) at (-2, 5) {$r_1$};
-
-
-\node [circle, minimum size = 0.1, draw] (c2) at (0, 5.4) {$c_2$};
-%
-%\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
-\node [circle, draw] (r2) at (2, 5) {$r_2$};
-\node [minimum width = 3cm, minimum height = 1cm, rectangle, draw] (stack) at (0, 2) {\large Stack};
-
-\path
-	(r)
-        edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
-
-\path   (r2)
-	edge [bend right, dashed] node {} (stack);
-\path   (c2)
-	edge [bend right, dashed] node {} (stack);
-
-\path   (r1)
-	edge [] node {} (r2);
-
-\end{tikzpicture}
-\caption{Second derivative taken}
-\end{figure}
-\noindent
-As the number of derivative steps increases,
-the stack would increase:
-
-\begin{figure}[H]
-\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
-%\draw (-6,-6) grid (6,6);
-\node  [ circle ] (r) at (-6, 5) {$r$};
-
-%\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
-\node  [circle, minimum size = 0.1, ] (c1) at (-4, 5.4) {$c_1$};
-%
-%\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
-\node  [minimum size = 0.5, circle, ] (r1) at (-2, 5) {$r_1$};
-
-
-\node [circle, minimum size = 0.1, ] (c2) at (0, 5.4) {$c_2$};
-%
-%\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
-\node [circle, ] (r2) at (2, 5) {$r_2$};
-\node [minimum width = 4cm, minimum height = 2.5cm, rectangle, draw] (stack) at (0, 1) { \large Stack};
-
-\node [] (ldots) at (3.5, 5) {};
-%\node (6, 5) (rn) circle [radius = 0.5] {$r_n$};
-
-\node [minimum size = 0.5, circle, ] (rn) at (6, 5) {};
-
-\node (rldots) at ($(ldots)!.4!(rn)$) {\ldots};
-
-\path
-	(r)
-        edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
-
-\path   (rldots)
-	edge [bend left, dashed] node {} (stack);
-
-\path   (r1)
-	edge [] node {} (r2);
-
-\path   (r2)
-	edge [] node {} (ldots);
-\path   (ldots)
-	edge [bend left, dashed] node {} (stack);
-\path   (5.03, 4.9)
-	edge [bend left, dashed] node {} (stack);
-\end{tikzpicture}
-\caption{More derivatives taken}
-\end{figure}
-
-
-\begin{figure}[H]
-\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
-%\draw (-6,-6) grid (6,6);
-\node  [radius = 0.5, circle, draw] (r) at (-6, 5) {$r$};
-
-%\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
-\node  [circle, minimum size = 0.1, draw] (c1) at (-4, 5.4) {$c_1$};
-%
-%\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
-\node  [minimum size = 0.5, circle, draw] (r1) at (-2, 5) {$r_1$};
-%
-%\node (0, 6)  (c2) circle [radius = 0.3] {$c_2$};
-\node [circle, minimum size = 0.1, draw] (c2) at (0, 5.4) {$c_2$};
-%
-%\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
-\node [circle, draw] (r2) at (2, 5) {$r_2$};
-%
-%
-\node [] (ldots) at (4.5, 5) {};
-%\node (6, 5) (rn) circle [radius = 0.5] {$r_n$};
-
-\node [minimum size = 0.5, circle, draw] (rn) at (6, 5) {$r_n$};
-
-\node at ($(ldots)!.4!(rn)$) {\ldots};
 
 
 
 
-\node [minimum size = 6cm, rectangle, draw] (stack) at (0, 0) {\Huge Stack};
-
-\path
-	(r)
-        edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
-\path
-	(r1)
-        edge [] node {} (r2);
-\path   (r2)
-	edge [] node {} (ldots);
-\path   (r)
-	edge [dashed, bend right] node {} (stack);
-
-\path   (r1)
-	edge [dashed, ] node {} (stack);
+% The first problem with this algorithm is that
+% for the function $\inj$ to work properly
+% we cannot destroy the structure of a regular expression,
+% and therefore cannot simplify along the way without mechanisms
+% that restores the values during the simplification process.
+% %too aggressively.
+% For example,
+% \[
+% 	r + (r + a) \rightarrow r + a
+% \]
+% cannot be applied because that would require
+% breaking up the inner alternative.
+% The $\lexer$ plus $\textit{simp}$ therefore only enables
+% same-level de-duplications like
+% \[
+% 	r + r \rightarrow r.
+% \]
+% Secondly, the algorithm recursively calls $\lexer$ on 
+% each new character input,
+% and before starting a child call
+% it stores information of previous lexing steps
+% on a stack, in the form of regular expressions
+% and characters: $r_0$, $c_0$, $r_1$, $c_1$, etc.
+% Each descent into deeper recursive calls in $\lexer$
+% causes a new pair of $r_i, c_i$ to be pushed to the call stack.
+% \begin{figure}[H]
+% \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
+% %\draw (-6,-6) grid (6,6);
+% \node  [ circle ] (r) at (-6, 5) {$r$};
 
-\path   (c1)
-	edge [dashed, bend right] node {} (stack);
+% %\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
+% \node  [circle, minimum size = 0.1, draw] (c1) at (-4, 5.4) {$c_1$};
+% %
+% %\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
+% \node  [minimum size = 0.5, circle, draw] (r1) at (-2, 5) {$r_1$};
+
+% \node [minimum width = 2cm, rectangle, draw] (stack) at (0, 3) {Stack};
+
+% \path
+% 	(r)
+%         edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
+
+% \path   (r1)
+% 	edge [bend right, dashed] node {saved} (stack);
+% \path   (c1)
+% 	edge [bend right, dashed] node {} (stack);
 
-\path   (c2)
-	edge [dashed] node {} (stack);
-\path   (4.5, 5)
-	edge [dashed, bend left] node {} (stack);
-\path   (4.9, 5)
-	edge [dashed, bend left] node {} (stack);
-\path   (5.3, 5)
-	edge [dashed, bend left] node {} (stack);
-\path (r2)
-	edge [dashed, ] node {} (stack);
-\path (rn)
-	edge [dashed, bend left] node {} (stack);
-\end{tikzpicture}
-\caption{Before injection phase starts}
-\end{figure}
+
+% \end{tikzpicture}
+% \caption{First derivative taken}
+% \end{figure}
+
+
+
+% \begin{figure}[H]
+% \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
+% %\draw (-6,-6) grid (6,6);
+% \node  [ circle ] (r) at (-6, 5) {$r$};
+
+% %\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
+% \node  [circle, minimum size = 0.1, ] (c1) at (-4, 5.4) {$c_1$};
+% %
+% %\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
+% \node  [minimum size = 0.5, circle, ] (r1) at (-2, 5) {$r_1$};
 
 
-\noindent
+% \node [circle, minimum size = 0.1, draw] (c2) at (0, 5.4) {$c_2$};
+% %
+% %\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
+% \node [circle, draw] (r2) at (2, 5) {$r_2$};
+% \node [minimum width = 3cm, minimum height = 1cm, rectangle, draw] (stack) at (0, 2) {\large Stack};
+
+% \path
+% 	(r)
+%         edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
+
+% \path   (r2)
+% 	edge [bend right, dashed] node {} (stack);
+% \path   (c2)
+% 	edge [bend right, dashed] node {} (stack);
+
+% \path   (r1)
+% 	edge [] node {} (r2);
+
+% \end{tikzpicture}
+% \caption{Second derivative taken}
+% \end{figure}
+% \noindent
+% As the number of derivative steps increases,
+% the stack would increase:
+
+% \begin{figure}[H]
+% \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
+% %\draw (-6,-6) grid (6,6);
+% \node  [ circle ] (r) at (-6, 5) {$r$};
+
+% %\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
+% \node  [circle, minimum size = 0.1, ] (c1) at (-4, 5.4) {$c_1$};
+% %
+% %\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
+% \node  [minimum size = 0.5, circle, ] (r1) at (-2, 5) {$r_1$};
+
+
+% \node [circle, minimum size = 0.1, ] (c2) at (0, 5.4) {$c_2$};
+% %
+% %\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
+% \node [circle, ] (r2) at (2, 5) {$r_2$};
+% \node [minimum width = 4cm, minimum height = 2.5cm, rectangle, draw] (stack) at (0, 1) { \large Stack};
+
+% \node [] (ldots) at (3.5, 5) {};
+% %\node (6, 5) (rn) circle [radius = 0.5] {$r_n$};
+
+% \node [minimum size = 0.5, circle, ] (rn) at (6, 5) {};
+
+% \node (rldots) at ($(ldots)!.4!(rn)$) {\ldots};
+
+% \path
+% 	(r)
+%         edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
+
+% \path   (rldots)
+% 	edge [bend left, dashed] node {} (stack);
+
+% \path   (r1)
+% 	edge [] node {} (r2);
+
+% \path   (r2)
+% 	edge [] node {} (ldots);
+% \path   (ldots)
+% 	edge [bend left, dashed] node {} (stack);
+% \path   (5.03, 4.9)
+% 	edge [bend left, dashed] node {} (stack);
+% \end{tikzpicture}
+% \caption{More derivatives taken}
+% \end{figure}
+
+
+% \begin{figure}[H]
+% \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
+% %\draw (-6,-6) grid (6,6);
+% \node  [radius = 0.5, circle, draw] (r) at (-6, 5) {$r$};
+
+% %\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
+% \node  [circle, minimum size = 0.1, draw] (c1) at (-4, 5.4) {$c_1$};
+% %
+% %\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
+% \node  [minimum size = 0.5, circle, draw] (r1) at (-2, 5) {$r_1$};
+% %
+% %\node (0, 6)  (c2) circle [radius = 0.3] {$c_2$};
+% \node [circle, minimum size = 0.1, draw] (c2) at (0, 5.4) {$c_2$};
+% %
+% %\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
+% \node [circle, draw] (r2) at (2, 5) {$r_2$};
+% %
+% %
+% \node [] (ldots) at (4.5, 5) {};
+% %\node (6, 5) (rn) circle [radius = 0.5] {$r_n$};
+
+% \node [minimum size = 0.5, circle, draw] (rn) at (6, 5) {$r_n$};
+
+% \node at ($(ldots)!.4!(rn)$) {\ldots};
+
+
+
+
+% \node [minimum size = 6cm, rectangle, draw] (stack) at (0, 0) {\Huge Stack};
+
+% \path
+% 	(r)
+%         edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
+% \path
+% 	(r1)
+%         edge [] node {} (r2);
+% \path   (r2)
+% 	edge [] node {} (ldots);
+% \path   (r)
+% 	edge [dashed, bend right] node {} (stack);
+
+% \path   (r1)
+% 	edge [dashed, ] node {} (stack);
+
+% \path   (c1)
+% 	edge [dashed, bend right] node {} (stack);
+
+% \path   (c2)
+% 	edge [dashed] node {} (stack);
+% \path   (4.5, 5)
+% 	edge [dashed, bend left] node {} (stack);
+% \path   (4.9, 5)
+% 	edge [dashed, bend left] node {} (stack);
+% \path   (5.3, 5)
+% 	edge [dashed, bend left] node {} (stack);
+% \path (r2)
+% 	edge [dashed, ] node {} (stack);
+% \path (rn)
+% 	edge [dashed, bend left] node {} (stack);
+% \end{tikzpicture}
+% \caption{Before injection phase starts}
+% \end{figure}
+
+
+% \noindent
 After all derivatives have been taken, the stack grows to a maximum size
 and the pair of regular expressions and characters $r_i, c_{i+1}$ 
 are then popped out and used in the injection phase.
 
 
 
-\begin{figure}[H]
-\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
-%\draw (-6,-6) grid (6,6);
-\node  [radius = 0.5, circle, ] (r) at (-6, 5) {$r$};
+% \begin{figure}[H]
+% \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,thick] 
+% %\draw (-6,-6) grid (6,6);
+% \node  [radius = 0.5, circle, ] (r) at (-6, 5) {$r$};
 
-%\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
-\node  [circle, minimum size = 0.1, ] (c1) at (-4, 5.4) {$c_1$};
-%
-%\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
-\node  [minimum size = 0.5, circle, ] (r1) at (-2, 5) {$r_1$};
-%
-%\node (0, 6)  (c2) circle [radius = 0.3] {$c_2$};
-\node [circle, minimum size = 0.1, ] (c2) at (0, 5.4) {$c_2$};
-%
-%\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
-\node [circle, ] (r2) at (2, 5) {$r_2$};
-%
-%
-\node [] (ldots) at (4.5, 5) {};
-%\node (6, 5) (rn) circle [radius = 0.5] {$r_n$};
+% %\node (-4, 6) (c1) circle [radius = 0.3] {$c_1$};
+% \node  [circle, minimum size = 0.1, ] (c1) at (-4, 5.4) {$c_1$};
+% %
+% %\node (-2, 5) (r1) circle [radius = 0.5] {$r_1$};
+% \node  [minimum size = 0.5, circle, ] (r1) at (-2, 5) {$r_1$};
+% %
+% %\node (0, 6)  (c2) circle [radius = 0.3] {$c_2$};
+% \node [circle, minimum size = 0.1, ] (c2) at (0, 5.4) {$c_2$};
+% %
+% %\node (2, 5) (r2) circle [radius = 0.5] {$r_2$};
+% \node [circle, ] (r2) at (2, 5) {$r_2$};
+% %
+% %
+% \node [] (ldots) at (4.5, 5) {};
+% %\node (6, 5) (rn) circle [radius = 0.5] {$r_n$};
 
-\node [minimum size = 0.5, circle, ] (rn) at (6, 5) {$r_n$};
+% \node [minimum size = 0.5, circle, ] (rn) at (6, 5) {$r_n$};
 
-\node at ($(ldots)!.4!(rn)$) {\ldots};
+% \node at ($(ldots)!.4!(rn)$) {\ldots};
 
-\node [minimum size = 0.5, circle, ] (vn) at (6, -5) {$v_n$};
+% \node [minimum size = 0.5, circle, ] (vn) at (6, -5) {$v_n$};
 
-\node [] (ldots2) at (3.5, -5) {};
+% \node [] (ldots2) at (3.5, -5) {};
 
-\node  [minimum size = 0.5, circle, ] (v2) at (2, -5) {$v_2$};
+% \node  [minimum size = 0.5, circle, ] (v2) at (2, -5) {$v_2$};
 
-\node at ($(ldots2)!.4!(v2)$) {\ldots};
+% \node at ($(ldots2)!.4!(v2)$) {\ldots};
 
 
-\node [circle, ] (v1) at (-2, -5) {$v_1$};
+% \node [circle, ] (v1) at (-2, -5) {$v_1$};
 
-\node  [radius = 0.5, circle, ] (v) at (-6, -5) {$v$};
+% \node  [radius = 0.5, circle, ] (v) at (-6, -5) {$v$};
 
-\node [minimum size = 6cm, rectangle, draw] (stack) at (0, 0) {\Huge Stack};
+% \node [minimum size = 6cm, rectangle, draw] (stack) at (0, 0) {\Huge Stack};
 
-\path
-	(r)
-        edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
-\path
-	(r1)
-        edge [] node {} (r2);
-\path   (r2)
-	edge [] node {} (ldots);
-\path   (rn)
-	edge [] node {$\mkeps$} (vn);
-\path   (vn) 
-	edge [] node {} (ldots2);
-\path   (v2)
-	edge [] node {$\inj \; r_1 \; c_2\;v_2$} (v1);
+% \path
+% 	(r)
+%         edge [->, >=stealth',shorten >=1pt] node[left] {} (r1);
+% \path
+% 	(r1)
+%         edge [] node {} (r2);
+% \path   (r2)
+% 	edge [] node {} (ldots);
+% \path   (rn)
+% 	edge [] node {$\mkeps$} (vn);
+% \path   (vn) 
+% 	edge [] node {} (ldots2);
+% \path   (v2)
+% 	edge [] node {$\inj \; r_1 \; c_2\;v_2$} (v1);
 
-\path   (v1)
-	edge [] node {$\inj \; r \; c_1 \; v_1$} (v);
+% \path   (v1)
+% 	edge [] node {$\inj \; r \; c_1 \; v_1$} (v);
 
-\path (stack)
-	edge [dashed] node {} (-4.2, -5.2);
-\path (stack)
-	edge [dashed] node {} (-4, -5.2);
-\path (stack)
-	edge [dashed] node {} (-0.1, -5.2);
-\path (stack)
-	edge [dashed] node {} (0.2, -5.26);
-\path (stack)
-	edge [dashed] node {} (3.2, -5);
-\path (stack)
-	edge [dashed] node {} (2.7, -5);
-\path (stack)
-	edge [dashed] node {} (3.7, -5);
-\end{tikzpicture}
-\caption{Stored $r_i, c_{i+1}$ Used by $\inj$}
-\end{figure}
-\noindent
+% \path (stack)
+% 	edge [dashed] node {} (-4.2, -5.2);
+% \path (stack)
+% 	edge [dashed] node {} (-4, -5.2);
+% \path (stack)
+% 	edge [dashed] node {} (-0.1, -5.2);
+% \path (stack)
+% 	edge [dashed] node {} (0.2, -5.26);
+% \path (stack)
+% 	edge [dashed] node {} (3.2, -5);
+% \path (stack)
+% 	edge [dashed] node {} (2.7, -5);
+% \path (stack)
+% 	edge [dashed] node {} (3.7, -5);
+% \end{tikzpicture}
+% \caption{Stored $r_i, c_{i+1}$ Used by $\inj$}
+% \end{figure}
+% \noindent
 Storing all $r_i, c_{i+1}$ pairs recursively
 allows the algorithm to work in an elegant way, at the expense of 
 storing quite a bit of verbose information on the stack.




