
diff -r ddd357703b6c -r d2c6852ca8da slides06.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/slides06.tex	Mon Oct 29 12:31:31 2012 +0000
@@ -0,0 +1,676 @@
+\documentclass[dvipsnames,14pt,t]{beamer}
+\usepackage{beamerthemeplainculight}
+\usepackage[T1]{fontenc}
+\usepackage[latin1]{inputenc}
+\usepackage{mathpartir}
+\usepackage[absolute,overlay]{textpos}
+\usepackage{ifthen}
+\usepackage{tikz}
+\usepackage{pgf}
+\usepackage{calc} 
+\usepackage{ulem}
+\usepackage{courier}
+\usepackage{listings}
+\renewcommand{\uline}[1]{#1}
+\usetikzlibrary{arrows}
+\usetikzlibrary{automata}
+\usetikzlibrary{shapes}
+\usetikzlibrary{shadows}
+\usetikzlibrary{positioning}
+\usetikzlibrary{calc}
+\usetikzlibrary{plotmarks}
+\usepackage{graphicx} 
+
+\definecolor{javared}{rgb}{0.6,0,0} % for strings
+\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
+\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
+\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
+
+\lstset{language=Java,
+	basicstyle=\ttfamily,
+	keywordstyle=\color{javapurple}\bfseries,
+	stringstyle=\color{javagreen},
+	commentstyle=\color{javagreen},
+	morecomment=[s][\color{javadocblue}]{/**}{*/},
+	numbers=left,
+	numberstyle=\tiny\color{black},
+	stepnumber=1,
+	numbersep=10pt,
+	tabsize=2,
+	showspaces=false,
+	showstringspaces=false}
+
+\lstdefinelanguage{scala}{
+  morekeywords={abstract,case,catch,class,def,%
+    do,else,extends,false,final,finally,%
+    for,if,implicit,import,match,mixin,%
+    new,null,object,override,package,%
+    private,protected,requires,return,sealed,%
+    super,this,throw,trait,true,try,%
+    type,val,var,while,with,yield},
+  otherkeywords={=>,<-,<\%,<:,>:,\#,@},
+  sensitive=true,
+  morecomment=[l]{//},
+  morecomment=[n]{/*}{*/},
+  morestring=[b]",
+  morestring=[b]',
+  morestring=[b]"""
+}
+
+\lstset{language=Scala,
+	basicstyle=\ttfamily,
+	keywordstyle=\color{javapurple}\bfseries,
+	stringstyle=\color{javagreen},
+	commentstyle=\color{javagreen},
+	morecomment=[s][\color{javadocblue}]{/**}{*/},
+	numbers=left,
+	numberstyle=\tiny\color{black},
+	stepnumber=1,
+	numbersep=10pt,
+	tabsize=2,
+	showspaces=false,
+	showstringspaces=false}
+
+% beamer stuff 
+\renewcommand{\slidecaption}{AFL 06, King's College London, 31.~October 2012}
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
+
+
+% The data files, written on the first run.
+\begin{filecontents}{re-python.data}
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+#29 58.886
+\end{filecontents}
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+\begin{filecontents}{re-internal.data}
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+10001 4.54569
+\end{filecontents}
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}<1>[t]
+\frametitle{%
+  \begin{tabular}{@ {}c@ {}}
+  \\[-3mm]
+  \LARGE Automata and \\[-2mm] 
+  \LARGE Formal Languages (6)\\[3mm] 
+  \end{tabular}}
+
+  \normalsize
+  \begin{center}
+  \begin{tabular}{ll}
+  Email:  & christian.urban at kcl.ac.uk\\
+  Of$\!$fice: & S1.27 (1st floor Strand Building)\\
+  Slides: & KEATS (also home work is there)\\
+  \end{tabular}
+  \end{center}
+
+
+\end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}ReDoS\end{tabular}}
+
+\begin{itemize}
+\item \alert{R}egular \alert{e}xpression \alert{D}enial \alert{o}f \alert{S}ervice\bigskip
+\item ``Regular Expressions Will Stab You in the Back''\bigskip
+\item Evil regular expressions\medskip
+\begin{itemize}
+\item \bl{$(a?\{n\})a\{n\}$}
+\item \bl{$(a^+)^+$}
+\item \bl{$([a-zA-Z]^+)^*$}
+\item \bl{$(a + aa)^+$}
+\item \bl{$(a + a?)^+$}
+\end{itemize}
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Regexp Matching\end{tabular}}
+
+Given a regular expression
+
+\begin{enumerate}
+\item you might convert it into a DFA (subset construction)
+\item you might try all possible paths in an NFA via backtracking
+\item you might try all paths in an NFA in parallel
+\item you might try to convert the DFA ``lazily''
+\end{enumerate}\bigskip
+
+Often No~2 is implemented (sometimes there are even good reasons for doing this).
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}\bl{$(a?\{n\})a\{n\}$} in Python\end{tabular}}
+
+\begin{tikzpicture}[y=.2cm, x=.3cm]
+ 	%axis
+	\draw (0,0) -- coordinate (x axis mid) (30,0);
+    	\draw (0,0) -- coordinate (y axis mid) (0,30);
+    	%ticks
+    	\foreach \x in {0,5,...,30}
+     		\draw (\x,1pt) -- (\x,-3pt)
+			node[anchor=north] {\x};
+    	\foreach \y in {0,5,...,30}
+     		\draw (1pt,\y) -- (-3pt,\y) 
+     			node[anchor=east] {\y}; 
+	%labels      
+	\node[below=0.6cm] at (x axis mid) {\bl{a}s};
+	\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
+
+	%plots
+	\draw[color=blue] plot[mark=*, mark options={fill=white}] 
+		file {re-python.data};
+	\only<2->{
+	\draw[color=red] plot[mark=triangle*, mark options={fill=white} ] 
+		file {re1.data};}
+	\only<3->{	 
+         \draw[color=green] plot[mark=square*, mark options={fill=white} ] 
+		file {re2.data};}
+    
+	%legend
+	\begin{scope}[shift={(4,20)}] 
+	\draw[color=blue] (0,0) -- 
+		plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0) 
+		node[right]{\small Python};
+	\only<2->{\draw[yshift=\baselineskip, color=red] (0,0) -- 
+		plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
+		node[right]{\small Scala V1};}
+	\only<3->{	
+	 \draw[yshift=2\baselineskip, color=green] (0,0) -- 
+		plot[mark=square*, mark options={fill=white}] (0.25,0) -- (0.5,0)
+		node[right]{\small Scala V2 with simplifications};}
+	\end{scope}
+\end{tikzpicture}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+
+\begin{tikzpicture}[y=.7cm, x=.0009cm]
+ 	%axis
+	\draw (0,0) -- coordinate (x axis mid) (10000,0);
+    	\draw (0,0) -- coordinate (y axis mid) (0,6);
+    	%ticks
+    	\foreach \x in {0,2000,...,10000}
+     		\draw (\x,1pt) -- (\x,-3pt)
+			node[anchor=north] {\x};
+    	\foreach \y in {0,1,...,6}
+     		\draw (1pt,\y) -- (-3pt,\y) 
+     			node[anchor=east] {\y}; 
+	%labels      
+	\node[below=0.6cm] at (x axis mid) {\bl{a}s};
+	\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
+
+	%plots
+	\draw[color=blue] plot[mark=*, mark options={fill=white}] 
+		file {re-internal.data};
+	\only<2->{	 
+         \draw[color=red] plot[mark=triangle*, mark options={fill=white} ] 
+		file {re3.data};}
+    
+	%legend
+	\begin{scope}[shift={(2000,4)}] 
+	\draw[color=blue] (0,0) -- 
+		plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0) 
+		node[right]{\small Scala Internal};
+        \only<2->{
+	\draw[yshift=\baselineskip, color=red] (0,0) -- 
+		plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
+		node[right]{\small Scala V3 with explicit $\_\{\_\}$};}
+	\end{scope}
+\end{tikzpicture}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
+
+Last week I showed you\bigskip
+
+\begin{itemize}
+\item an algorithm for automata minimisation
+
+\item an algorithm for transforming a regular expression into an NFA
+
+\item an algorithm for transforming an NFA into a DFA (subset construction)
+
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}This Week\end{tabular}}
+
+Go over the algorithms again, but with two new things and \ldots\medskip
+
+\begin{itemize}
+\item with the example: what is the regular expression that accepts every string, except those ending 
+in \bl{aa}?\medskip
+
+\item Go over the proof for \bl{$L(rev(r)) = Rev(L(r))$}.\medskip
+
+\item Anything else so far.
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs By Induction\end{tabular}}
+
+\begin{itemize}
+\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
+\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} already
+holds for \bl{r$_1$} and \bl{r$_2$}.\bigskip
+\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} already
+holds for \bl{r$_1$} and \bl{r$_2$}.
+\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} already
+holds for \bl{r}.
+\end{itemize}
+
+\begin{center}
+\bl{$P(r):\;\;L(rev(r)) = Rev(L(r))$}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+
+What is the regular expression that accepts every string, except those ending 
+in \bl{aa}?\pause\bigskip
+
+\begin{center}
+\begin{tabular}{l}
+\bl{(a + b)$^*$ba}\\
+\bl{(a + b)$^*$ab}\\
+\bl{(a + b)$^*$bb}\\\pause
+\bl{a}\\
+\bl{\texttt{""}}
+\end{tabular}
+\end{center}\pause
+
+What are the strings to be avoided?\pause\medskip
+
+\begin{center}
+\bl{(a + b)$^*$aa}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+
+An NFA for \bl{(a + b)$^*$aa}
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+  \node[state, initial]        (q0) at ( 0,1) {$q_0$};
+  \node[state]                    (q1) at ( 1,1) {$q_1$};
+  \node[state, accepting] (q2) at ( 2,1) {$q_2$};
+  \path[->] (q0) edge node[above] {$a$} (q1)
+                  (q1) edge node[above] {$a$} (q2)
+                  (q0) edge [loop below] node {$a$} ()
+                  (q0) edge [loop above] node {$b$} ()
+            ;
+\end{tikzpicture}
+\end{center}\pause
+
+Minimisation for DFAs\\
+Subset Construction for NFAs
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}DFA Minimisation\end{tabular}}
+
+
+\begin{enumerate}
+\item Take all pairs \bl{(q, p)} with \bl{q $\not=$ p}
+\item Mark all pairs that accepting and non-accepting states
+\item For  all unmarked pairs \bl{(q, p)} and all characters \bl{c} tests wether
+\begin{center}
+\bl{($\delta$(q,c), $\delta$(p,c))}
+\end{center} 
+are marked. If yes, then also mark \bl{(q, p)}.
+\item Repeat last step until nothing changed.
+\item All unmarked pairs can be merged.
+\end{enumerate}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+Minimal DFA \only<1>{\bl{(a + b)$^*$aa}}\only<2->{\alert{not} \bl{(a + b)$^*$aa}}
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+  \only<1>{\node[state, initial]        (q0) at ( 0,1) {$q_0$};}
+  \only<2->{\node[state, initial,accepting]        (q0) at ( 0,1) {$q_0$};}
+  \only<1>{\node[state]                    (q1) at ( 1,1) {$q_1$};}
+  \only<2->{\node[state,accepting]                    (q1) at ( 1,1) {$q_1$};}
+  \only<1>{\node[state, accepting] (q2) at ( 2,1) {$q_2$};}
+  \only<2->{\node[state] (q2) at ( 2,1) {$q_2$};}
+  \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
+                  (q1) edge[bend left] node[above] {$b$} (q0)
+                  (q2) edge[bend left=50] node[below] {$b$} (q0)
+                  (q1) edge node[above] {$a$} (q2)
+                  (q2) edge [loop right] node {$a$} ()
+                  (q0) edge [loop below] node {$b$} ()
+            ;
+\end{tikzpicture}
+\end{center}
+
+\onslide<3>{How to get from a DFA to a regular expression?}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+  \only<1->{\node[state, initial]        (q0) at ( 0,1) {$q_0$};}
+  \only<1->{\node[state]                    (q1) at ( 1,1) {$q_1$};}
+  \only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
+  \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
+                  (q1) edge[bend left] node[above] {$b$} (q0)
+                  (q2) edge[bend left=50] node[below] {$b$} (q0)
+                  (q1) edge node[above] {$a$} (q2)
+                  (q2) edge [loop right] node {$a$} ()
+                  (q0) edge [loop below] node {$b$} ()
+            ;
+\end{tikzpicture}
+\end{center}\pause\bigskip
+
+\onslide<2->{
+\begin{center}
+\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
+\bl{$q_0$} & \bl{$=$} & \bl{$2\, q_0 + 3 \,q_1 +  4\, q_2$}\\
+\bl{$q_1$} & \bl{$=$} & \bl{$2 \,q_0 + 3\, q_1 + 1\, q_2$}\\
+\bl{$q_2$} & \bl{$=$} & \bl{$1\, q_0 + 5\, q_1 + 2\, q_2$}\\
+
+\end{tabular}
+\end{center}
+}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+  \only<1->{\node[state, initial]        (q0) at ( 0,1) {$q_0$};}
+  \only<1->{\node[state]                    (q1) at ( 1,1) {$q_1$};}
+  \only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
+  \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
+                  (q1) edge[bend left] node[above] {$b$} (q0)
+                  (q2) edge[bend left=50] node[below] {$b$} (q0)
+                  (q1) edge node[above] {$a$} (q2)
+                  (q2) edge [loop right] node {$a$} ()
+                  (q0) edge [loop below] node {$b$} ()
+            ;
+\end{tikzpicture}
+\end{center}\bigskip
+
+\onslide<2->{
+\begin{center}
+\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
+\bl{$q_0$} & \bl{$=$} & \bl{$\epsilon + q_0\,b + q_1\,b +  q_2\,b$}\\
+\bl{$q_1$} & \bl{$=$} & \bl{$q_0\,a$}\\
+\bl{$q_2$} & \bl{$=$} & \bl{$q_1\,a + q_2\,a$}\\
+
+\end{tabular}
+\end{center}
+}
+
+\onslide<3->{
+Arden's Lemma:
+\begin{center}
+If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
+\end{center}
+}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Algorithms on Automata\end{tabular}}
+
+
+\begin{itemize}
+\item Reg $\rightarrow$ NFA: Thompson-McNaughton-Yamada method\medskip
+\item NFA $\rightarrow$ DFA: Subset Construction\medskip
+\item DFA $\rightarrow$ Reg: Brzozowski's Algebraic Method\medskip
+\item DFA minimisation: Hopcrofts Algorithm\medskip
+\item complement DFA
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+\newcommand{\qq}{\mbox{\texttt{"}}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Grammars\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
+$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
+$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
+\end{tabular}}
+\end{center}
+
+\bl{$E$}, \bl{$F$} and \bl{$T$} are non-terminals\\
+\bl{$E$} is start symbol\\
+\bl{$num$}, \bl{(}, \bl{)}, \bl{+} \ldots are terminals\bigskip\\
+
+
+\bl{\texttt{(2*3)+(3+4)}}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
+$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
+$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
+\end{tabular}}
+\end{center}
+
+\begin{center}
+\begin{tikzpicture}[level distance=8mm, blue]
+  \node {E}
+    child {node {F} 
+     child {node {T} 
+                 child {node {\qq(\qq\,E\,\qq)\qq}
+                            child {node{F \qq*\qq{} F}
+                                  child {node {T} child {node {2}}}
+                                  child {node {T} child {node {3}}} 
+                               }
+                          }
+              }
+     child {node {\qq+\qq}}
+     child {node {T}
+       child {node {\qq(\qq\,E\,\qq)\qq} 
+       child {node {F}
+       child {node {T \qq+\qq{} T}
+                    child {node {3}}
+                    child {node {4}} 
+                 }
+                 }}
+    }};
+\end{tikzpicture}
+\end{center}
+
+\begin{textblock}{5}(1, 5)
+\bl{\texttt{(2*3)+(3+4)}}
+\end{textblock}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+\end{document}
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% End: 
+




