
diff -r e60c4a9ba340 -r 57df3d7b4a25 slides/slides06.tex
--- a/slides/slides06.tex	Tue Oct 29 12:24:22 2013 +0000
+++ b/slides/slides06.tex	Wed Oct 30 03:36:10 2013 +0000
@@ -82,120 +82,52 @@
 \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}
-1 0.029
-5 0.029
-10 0.029
-15 0.032
-16 0.042
-17 0.042
-18 0.055
-19 0.084
-20 0.136
-21 0.248
-22 0.464
-23 0.899
-24 1.773
-25 3.505
-26 6.993
-27 14.503
-28 29.307
-#29 58.886
-\end{filecontents}
 
-\begin{filecontents}{re1.data}
-1 0.00179
-2 0.00011
-3 0.00014
-4 0.00026
-5 0.00050
-6 0.00095
-7 0.00190
-8 0.00287
-9 0.00779
-10 0.01399
-11 0.01894
-12 0.03666
-13 0.07994
-14 0.08944
-15 0.02377
-16 0.07392
-17 0.22798
-18 0.65310
-19 2.11360
-20 6.31606
-21 21.46013
+\begin{filecontents}{s-grammar1.data}
+1 0.01152
+51 0.07973
+101 0.09726
+151 0.09320
+201 0.10010
+251 0.16997
+301 0.26662
+351 0.46118
+401 0.62516
+451 0.87247
+501 1.16334
+551 1.71152
+601 2.10958
+651 2.44360
+701 2.98488
+751 3.50326
+801 4.11036
+851 4.93394
+901 5.77465
+951 7.39123
 \end{filecontents}
 
-\begin{filecontents}{re2.data}
-1  0.00240
-2  0.00013
-3  0.00020
-4  0.00030
-5  0.00049
-6  0.00083
-7  0.00146
-8  0.00228
-9  0.00351
-10  0.00640
-11  0.01217
-12  0.02565
-13  0.01382
-14  0.02423
-15  0.05065 
-16  0.06522
-17  0.02140
-18  0.05176
-19  0.18254
-20  0.51898
-21  1.39631
-22  2.69501
-23  8.07952
+\begin{filecontents}{s-grammar2.data}
+1 0.01280
+2 0.00064
+3 0.00173
+4 0.00355
+5 0.00965
+6 0.02674
+7 0.06953
+8 0.11166
+9 0.18707
+10 0.09189
+11 0.12724
+12 0.24337
+13 0.59304
+14 1.53594
+15 4.01195
+16 10.73582
+17 29.51587
+#18 73.14163
 \end{filecontents}
 
-\begin{filecontents}{re-internal.data}
-1 0.00069
-301 0.00700
-601 0.00297
-901 0.00470
-1201 0.01301
-1501 0.01175
-1801 0.01761
-2101 0.01787
-2401 0.02717
-2701 0.03932
-3001 0.03470
-3301 0.04349
-3601 0.05411
-3901 0.06181
-4201 0.07119
-4501 0.08578
-\end{filecontents}
 
-\begin{filecontents}{re3.data}
-1 0.001605
-501 0.131066
-1001 0.057885
-1501 0.136875
-2001 0.176238
-2501 0.254363
-3001 0.37262
-3501 0.500946
-4001 0.638384
-4501 0.816605
-5001 1.00491
-5501 1.232505
-6001 1.525672
-6501 1.757502
-7001 2.092784
-7501 2.429224
-8001 2.803037
-8501 3.463045
-9001 3.609
-9501 4.081504
-10001 4.54569
-\end{filecontents}
 \begin{document}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -239,9 +171,10 @@
 \bl{$A \rightarrow \text{rhs}$}
 \end{center}
 
-where \bl{rhs} are sequences involving terminals and nonterminals.\medskip\pause
+where \bl{rhs} are sequences involving terminals and nonterminals,
+including the empty sequence \bl{$\epsilon$}.\medskip\pause
 
-We can also allow rules
+We also allow rules
 \begin{center}
 \bl{$A \rightarrow \text{rhs}_1 | \text{rhs}_2 | \ldots$}
 \end{center}
@@ -299,6 +232,111 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}A CFG Derivation\end{tabular}}
+
+\begin{enumerate}
+\item Begin with a string containing only the start symbol, say \bl{$S$}\bigskip
+\item Replace any nonterminal \bl{$X$} in the string by the
right-hand side of some production \bl{$X \rightarrow \text{rhs}$}\bigskip
+\item Repeat 2 until there are no nonterminals
+\end{enumerate}
+
+\begin{center}
+\bl{$S \rightarrow \ldots \rightarrow \ldots  \rightarrow \ldots  \rightarrow \ldots $}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Example Derivation\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$S$ & $\rightarrow$ &  $\epsilon \;|\; a\cdot S\cdot a \;|\;b\cdot S\cdot b$ \\
+\end{tabular}}
+\end{center}\bigskip
+
+
+\begin{center}
+\begin{tabular}{lcl}
+\bl{$S$} & \bl{$\rightarrow$} & \bl{$aSa$}\\
+              & \bl{$\rightarrow$} & \bl{$abSba$}\\
+              & \bl{$\rightarrow$} & \bl{$abaSaba$}\\
+              & \bl{$\rightarrow$} & \bl{$abaaba$}\\
+
+ 
+\end{tabular}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Example Derivation\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $num\_token$ \\
+$E$ & $\rightarrow$ &  $E \cdot + \cdot E$ \\
+$E$ & $\rightarrow$ &  $E \cdot - \cdot E$ \\
+$E$ & $\rightarrow$ &  $E \cdot * \cdot E$ \\
+$E$ & $\rightarrow$ &  $( \cdot E \cdot )$ 
+\end{tabular}}
+\end{center}\bigskip
+
+
+\begin{center}
+\begin{tabular}{@{}c@{}c@{}}
+\begin{tabular}{l@{\hspace{1mm}}l@{\hspace{1mm}}l}
+\bl{$E$} & \bl{$\rightarrow$} & \bl{$E*E$}\\
+              & \bl{$\rightarrow$} & \bl{$E+E*E$}\\
+              & \bl{$\rightarrow$} & \bl{$E+E*E+E$}\\
+              & \bl{$\rightarrow^+$} & \bl{$1+2*3+4$}\\
+\end{tabular} &\pause
+\begin{tabular}{l@{\hspace{1mm}}l@{\hspace{1mm}}l}
+\bl{$E$} & \bl{$\rightarrow$} & \bl{$E+E$}\\
+              & \bl{$\rightarrow$} & \bl{$E+E+E$}\\
+              & \bl{$\rightarrow$} & \bl{$E+E*E+E$}\\
+              & \bl{$\rightarrow^+$} & \bl{$1+2*3+4$}\\
+\end{tabular}
+\end{tabular}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Language of a CFG\end{tabular}}
+
+Let \bl{$G$} be a context-free grammar with start symbol \bl{$S$}. 
+Then the language \bl{$L(G)$} is:
+
+\begin{center}
+\bl{$\{c_1\ldots c_n \;|\; \forall i.\; c_i \in T \wedge S \rightarrow^* c_1\ldots c_n \}$}
+\end{center}\pause
+
+\begin{itemize}
+\item Terminals, because there are no rules for replacing them.
+\item Once generated, terminals are ``permanent''.
+\item Terminals ought to be tokens of the language\\
+(but can also be strings).
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
@@ -343,12 +381,36 @@
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Arithmetic Expressions\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $num\_token$ \\
+$E$ & $\rightarrow$ &  $E \cdot + \cdot E$ \\
+$E$ & $\rightarrow$ &  $E \cdot - \cdot E$ \\
+$E$ & $\rightarrow$ &  $E \cdot * \cdot E$ \\
+$E$ & $\rightarrow$ &  $( \cdot E \cdot )$ 
+\end{tabular}}
+\end{center}\pause\bigskip
+
+A CFG is \alert{left-recursive} if it has a nonterminal \bl{$E$} such
+that \bl{$E \rightarrow^+ E\cdot \ldots$}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
 \frametitle{\begin{tabular}{c}Ambiguous Grammars\end{tabular}}
 
-A grammar is \alert{ambiguous} if there is a string that has at least parse trees.
+A grammar is \alert{ambiguous} if there is a string that has at least two different parse trees.
 
 
 \begin{center}
@@ -366,6 +428,362 @@
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Dangling Else\end{tabular}}
+
+Another ambiguous grammar:\bigskip
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  if $E$ then $E$\\
+ & $|$ &  if $E$ then $E$ else $E$ \\
+ & $|$ &  \ldots
+\end{tabular}}
+\end{center}\bigskip
+
+\bl{\texttt{if a then if x then y else c}}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Parser Combinators\end{tabular}}
+
+Parser combinators: \bigskip
+
+\begin{minipage}{1.1\textwidth}
+\begin{center}
+\mbox{}\hspace{-12mm}\mbox{}$\underbrace{\text{list of tokens}}_{\text{input}}$ \bl{$\Rightarrow$} 
+$\underbrace{\text{set of (parsed input, unparsed input)}}_{\text{output}}$
+\end{center}
+\end{minipage}\bigskip
+
+\begin{itemize}
+\item sequencing
+\item alternative
+\item semantic action
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+Alternative parser (code \bl{$p\;||\;q$})\bigskip
+
+\begin{itemize}
+\item apply \bl{$p$} and also \bl{$q$}; then combine the outputs
+\end{itemize}
+
+\begin{center}
+\large \bl{$p(\text{input}) \cup q(\text{input})$}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+Sequence parser (code \bl{$p\sim q$})\bigskip
+
+\begin{itemize}
+\item apply first \bl{$p$} producing a set of pairs
+\item then apply \bl{$q$} to the unparsed parts
+\item then combine the results:\\ \mbox{}\;\;((output$_1$, output$_2$), unparsed part)
+\end{itemize}
+
+\begin{center}
+\begin{tabular}{l}
+\large \bl{$\{((o_1, o_2), u_2) \;|\;$}\\[2mm] 
+\large\mbox{}\hspace{15mm} \bl{$(o_1, u_1) \in p(\text{input}) \wedge$}\\[2mm]
+\large\mbox{}\hspace{15mm} \bl{$(o_2, u_2) \in q(u_1)\}$}
+\end{tabular}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+Function parser (code \bl{$p \Rightarrow f$})\bigskip
+
+\begin{itemize}
+\item apply \bl{$p$} producing a set of pairs
+\item then apply the function \bl{$f$} to each first component
+\end{itemize}
+
+\begin{center}
+\begin{tabular}{l}
+\large \bl{$\{(f(o_1), u_1) \;|\; (o_1, u_1) \in p(\text{input})\}$}
+\end{tabular}
+\end{center}\bigskip\bigskip\pause
+
+\bl{$f$} is the semantic action (``what to do with the parsed input'')
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Semantic Actions\end{tabular}}
+
+Addition
+
+\begin{center}
+\bl{$T \sim + \sim E \Rightarrow \underbrace{f((x,y), z) \Rightarrow x + z}_{\text{semantic action}}$}
+\end{center}\pause
+
+Multiplication
+
+\begin{center}
+\bl{$F \sim * \sim T \Rightarrow f((x,y), z) \Rightarrow x * z$}
+\end{center}\pause
+
+Parenthesis
+
+\begin{center}
+\bl{$\text{(} \sim E \sim \text{)} \Rightarrow f((x,y), z) \Rightarrow y$}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Types of Parsers\end{tabular}}
+
+\begin{itemize}
+\item {\bf Sequencing}: if \bl{$p$} returns results of type \bl{$T$}, and \bl{$q$} results of type \bl{$S$},
+then \bl{$p \sim q$} returns results of type
+
+\begin{center}
+\bl{$T \times S$}
+\end{center}\pause
+
+\item {\bf Alternative}: if \bl{$p$} returns results of type \bl{$T$} then  \bl{$q$} \alert{must} also have results of type \bl{$T$},
+and \bl{$p \;||\; q$} returns results of type
+
+\begin{center}
+\bl{$T$}
+\end{center}\pause
+
+\item {\bf Semantic Action}: if \bl{$p$} returns results of type \bl{$T$} and \bl{$f$} is a function from
+\bl{$T$} to \bl{$S$}, then
+\bl{$p \Rightarrow f$} returns results of type
+
+\begin{center}
+\bl{$S$}
+\end{center}
+
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Input Types of Parsers\end{tabular}}
+
+\begin{itemize}
+\item input: \alert{string}
+\item output: set of (output\_type, \alert{string})
+\end{itemize}\bigskip\pause
+
+actually it can be any input type as long as it is a kind of sequence
+(for example a string)
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Scannerless Parsers\end{tabular}}
+
+\begin{itemize}
+\item input: \alert{string}
+\item output: set of (output\_type, \alert{string})
+\end{itemize}\bigskip
+
+but lexers are better when whitespaces or comments need to be filtered out;
+then input is a sequence of tokens
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Successful Parses\end{tabular}}
+
+\begin{itemize}
+\item input: string
+\item output: \alert{set of} (output\_type, string)
+\end{itemize}\bigskip
+
+a parse is successful whenever the input has been
+fully ``consumed'' (that is the second component is empty)
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Abstract Parsers}
+
+\mbox{\lstset{language=Scala}\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting{../progs/app7.scala}}}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+{\lstset{language=Scala}\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting{../progs/app8.scala}}}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Two Grammars\end{tabular}}
+
+Which languages are recognised by the following two grammars?
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$S$ & $\rightarrow$ &  $1 \cdot S \cdot S$\\
+        & $|$ & $\epsilon$
+\end{tabular}}
+\end{center}\bigskip
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$U$ & $\rightarrow$ &  $1 \cdot U$\\
+        & $|$ & $\epsilon$
+\end{tabular}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Ambiguous Grammars\end{tabular}}
+
+\mbox{}\\[-25mm]\mbox{}
+
+\begin{center}
+\begin{tikzpicture}[y=.2cm, x=.009cm]
+ 	%axis
+	\draw (0,0) -- coordinate (x axis mid) (1000,0);
+    	\draw (0,0) -- coordinate (y axis mid) (0,30);
+    	%ticks
+    	\foreach \x in {0, 20, 100, 200,...,1000}
+     		\draw (\x,1pt) -- (\x,-3pt)
+			node[anchor=north] {\small \x};
+    	\foreach \y in {0,5,...,30}
+     		\draw (1pt,\y) -- (-3pt,\y) 
+     			node[anchor=east] {\small\y}; 
+	%labels      
+	\node[below=0.6cm] at (x axis mid) {\bl{1}s};
+	\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
+
+	%plots
+	\draw[color=blue] plot[mark=*, mark options={fill=white}] 
+		file {s-grammar1.data};
+         \only<2->{\draw[color=red] plot[mark=triangle*, mark options={fill=white} ] 
+                  file {s-grammar2.data};}
+	%legend
+	\begin{scope}[shift={(400,20)}] 
+	\draw[color=blue] (0,0) -- 
+		plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0) 
+		node[right]{\small unambiguous};
+	\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 ambiguous};}  
+	\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}While-Language\end{tabular}}
+
+
+\begin{center}
+\bl{\begin{tabular}{@{}lcl@{}}
+$Stmt$ & $\rightarrow$ &  $\text{skip}$\\
+              & $|$ & $Id := AExp$\\
+              & $|$ & $\text{if}\; B\!Exp \;\text{then}\; Block \;\text{else}\; Block$\\
+              & $|$ & $\text{while}\; B\!Exp \;\text{do}\; Block$\medskip\\
+$Stmts$ & $\rightarrow$ &  $Stmt \;\text{;}\; Stmts$\\
+              & $|$ & $Stmt$\medskip\\
+$Block$ & $\rightarrow$ &  $\{ Stmts \}$\\
+                & $|$ & $Stmt$\medskip\\
+$AExp$ & $\rightarrow$ & \ldots\\
+$BExp$ & $\rightarrow$ & \ldots\\
+\end{tabular}}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}An Interpreter\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{l}
+$\{$\\
+\;\;$x := 5 \text{;}$\\
+\;\;$y := x * 3\text{;}$\\
+\;\;$y := x * 4\text{;}$\\
+\;\;$x := u * 3$\\
+$\}$
+\end{tabular}}
+\end{center}
+
+\begin{itemize}
+\item the interpreter has to record the value of \bl{$x$} before assigning a value to \bl{$y$}\pause
+\item \bl{\text{eval}(stmt, env)}
+\end{itemize}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{




