# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1446785681 0
# Node ID 9b71dead1219e61db2a18a47b60ab932f78db2d7
# Parent  50ce3667c19085f5f67df2f2b74457ddfad41089
updated

diff -r 50ce3667c190 -r 9b71dead1219 progs/pow.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/progs/pow.scala	Fri Nov 06 04:54:41 2015 +0000
@@ -0,0 +1,20 @@
+def concat(A: Set[String], B: Set[String]) : Set[String] =
+  for (x <-A ; y <- B) yield x ++ y
+
+def pow(A: Set[String], n: Int) : Set[String] = n match {
+  case 0 => Set("")
+  case n => concat(A, pow(A, n- 1))
+}
+
+val A = Set("a", "b", "c", "d")
+pow(A, 4).size                            // -> 256
+
+val B = Set("a", "b", "c", "")
+pow(B, 4).size                            // -> 121
+
+val C = Set("a", "b", "")
+pow(C, 2)
+pow(C, 2).size                            // -> 7
+
+pow(C, 3)
+pow(C, 3).size                            // -> 15
diff -r 50ce3667c190 -r 9b71dead1219 slides/slides05.pdf
Binary file slides/slides05.pdf has changed
diff -r 50ce3667c190 -r 9b71dead1219 slides/slides06.pdf
Binary file slides/slides06.pdf has changed
diff -r 50ce3667c190 -r 9b71dead1219 slides/slides06.tex
--- a/slides/slides06.tex	Sat Oct 31 11:37:55 2015 +0000
+++ b/slides/slides06.tex	Fri Nov 06 04:54:41 2015 +0000
@@ -3,21 +3,15 @@
 \usepackage{../graphics}
 \usepackage{../langs}
 \usepackage{../data}
-\usepackage{../grammar}
 
-\hfuzz=220pt 
-
-\pgfplotsset{compat=1.11}
-
-\newcommand{\bl}[1]{\textcolor{blue}{#1}}  
 
 % beamer stuff 
 \renewcommand{\slidecaption}{AFL 06, King's College London}
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
+%\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
 
 \begin{document}
 
-
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t]
 \frametitle{%
@@ -41,35 +35,98 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
-\frametitle{Regular Languages}
 
-While regular expressions are very useful for lexing, there is
-no regular expression that can recognise the language
-\bl{$a^nb^n$}.\bigskip
+\mbox{}\\[-18mm]\mbox{}
 
-\begin{center}
-\bl{$(((()()))())$} \;\;vs.\;\; \bl{$(((()()))()))$}
-\end{center}\bigskip\bigskip
-
-\small
-\noindent So we cannot find out with regular expressions
-whether parentheses are matched or unmatched.
+{\lstset{language=Scala}\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting[xleftmargin=0mm]{../progs/pow.scala}}}
 
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
-\frametitle{Hierarchy of Languages}
+\small
+\mbox{}\\[5mm]
+%\begin{textblock}{10}(3,5)
+\begin{tikzpicture}[scale=1.5,
+                    node distance=1cm,
+                    every node/.style={minimum size=7mm}]
+\node (r1)  {\bl{$r_1$}};
+\node (r2) [right=of r1] {\bl{$r_2$}};
+\draw[->,line width=1mm]  (r1) -- (r2) node[above,midway] {\bl{$der\,a$}};
+\node (r3) [right=of r2] {\bl{$r_3$}};
+\draw[->,line width=1mm]  (r2) -- (r3) node[above,midway] {\bl{$der\,b$}};
+\node (r4) [right=of r3] {\bl{$r_4$}};
+\draw[->,line width=1mm]  (r3) -- (r4) node[above,midway] {\bl{$der\,c$}};
+\draw (r4) node[anchor=west] {\;\raisebox{3mm}{\bl{$nullable$}}};
+\node (v4) [below=of r4] {\bl{$v_4$}};
+\draw[->,line width=1mm]  (r4) -- (v4);
+\node (v3) [left=of v4] {\bl{$v_3$}};
+\draw[->,line width=1mm]  (v4) -- (v3) node[below,midway] {\bl{$inj\,c$}};
+\node (v2) [left=of v3] {\bl{$v_2$}};
+\draw[->,line width=1mm]  (v3) -- (v2) node[below,midway] {\bl{$inj\,b$}};
+\node (v1) [left=of v2] {\bl{$v_1$}};
+\draw[->,line width=1mm]  (v2) -- (v1) node[below,midway] {\bl{$inj\,a$}};
+\draw[->,line width=0.5mm]  (r3) -- (v3);
+\draw[->,line width=0.5mm]  (r2) -- (v2);
+\draw[->,line width=0.5mm]  (r1) -- (v1);
+\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{\bl{$mkeps$}}};
+\end{tikzpicture}
+%\end{textblock}
+
+\begin{center}
+\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
+  \\[-10mm]
+  \bl{$inj\,(c)\,c\,Empty$} & \bl{$\dn$}  & \bl{$Char\,c$}\\
+  \bl{$inj\,(r_1 + r_2)\,c\,Left(v)$} & \bl{$\dn$}  & \bl{$Left(inj\,r_1\,c\,v)$}\\
+  \bl{$inj\,(r_1 + r_2)\,c\,Right(v)$} & \bl{$\dn$}  & \bl{$Right(inj\,r_2\,c\,v)$}\\
+  \bl{$inj\,(r_1 \cdot r_2)\,c\,Seq(v_1,v_2)$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\
+  \bl{$inj\,(r_1 \cdot r_2)\,c\,Left(Seq(v_1,v_2))$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\
+  \bl{$inj\,(r_1 \cdot r_2)\,c\,Right(v)$} & \bl{$\dn$}  & \bl{$Seq(mkeps(r_1),inj\,r_2\,c\,v)$}\\
+  \bl{$inj\,(r^*)\,c\,Seq(v,vs)$} & \bl{$\dn$}  & \bl{$inj\,r\,c\,v\,::\,vs$}\\
+\end{tabular}
+\end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+
+\newcommand{\qq}{\mbox{\texttt{"}}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{CFGs}
+
+A \alert{context-free} grammar (CFG) \bl{$G$} consists of:
+
+\begin{itemize}
+\item a finite set of nonterminal symbols (upper case)
+\item a finite terminal symbols or tokens (lower case)
+\item a start symbol (which must be a nonterminal)
+\item a set of rules
+\begin{center}
+\bl{$A \rightarrow \text{rhs}_1 | \text{rhs}_2 | \ldots$}
+\end{center}
+
+where \bl{rhs} are sequences involving terminals and nonterminals (can also be empty).\medskip\pause
+
+\end{itemize}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Hierarchy of Languages\end{tabular}}
+
+Recall that languages are sets of strings.
 
 \begin{center}
 \begin{tikzpicture}
-[rect/.style={draw=black!50, 
-              top color=white,
-              bottom color=black!20, 
-              rectangle, 
-              very thick, 
-              rounded corners}]
+[rect/.style={draw=black!50, top color=white,bottom color=black!20, rectangle, very thick, rounded corners}]
 
 \draw (0,0) node [rect, text depth=39mm, text width=68mm] {all languages};
 \draw (0,-0.4) node [rect, text depth=28.5mm, text width=64mm] {decidable languages};
@@ -80,367 +137,262 @@
 
 \end{center}
 
-\end{frame}
+\end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
 \begin{frame}[c]
-\frametitle{Grammars}
+\frametitle{\begin{tabular}{c}Arithmetic Expressions\end{tabular}}
+
+A grammar for arithmetic expressions and numbers:
 
-A (context-free) grammar \bl{$G$} consists of
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $E \cdot + \cdot E \;|\;E \cdot * \cdot E \;|\;( \cdot E \cdot ) \;|\;N$ \\
+$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1 \;|\: \ldots \;|\; 9$ 
+\end{tabular}}
+\end{center}
 
-\begin{itemize}
-\item a finite set of nonterminal symbols (upper case)
-\item a finite terminal symbols or tokens (lower case)
-\item a start symbol (which must be a nonterminal)
-\item a set of rules
+Unfortunately it is left-recursive (and ambiguous).\medskip\\
+A problem for \alert{recursive descent parsers} (e.g.~parser combinators).
+\bigskip\pause
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Numbers\end{tabular}}
+
+
+
 \begin{center}
-\bl{$A \rightarrow \text{rhs}$}
+\bl{\begin{tabular}{lcl}
+$N$ & $\rightarrow$ &  $N \cdot N \;|\; 0 \;|\; 1 \;|\; \ldots \;|\; 9$\\
+\end{tabular}}
 \end{center}
 
-where \bl{rhs} are sequences involving terminals and nonterminals,
-including the empty sequence \bl{$\epsilon$}.\medskip\pause
+A non-left-recursive, non-ambiguous grammar for numbers:
 
-We also allow rules
 \begin{center}
-\bl{$A \rightarrow \text{rhs}_1 | \text{rhs}_2 | \ldots$}
+\bl{\begin{tabular}{lcl}
+$N$ & $\rightarrow$ &  $0 \cdot N \;|\;1 \cdot N \;|\;\ldots\;|\; 0 \;|\; 1 \;|\; \ldots \;|\; 9$\\
+\end{tabular}}
 \end{center}
-\end{itemize}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Operator Precedences\end{tabular}}
+
+
+To disambiguate
 
-\end{frame}
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E$ & $\rightarrow$ &  $E \cdot + \cdot E \;|\;E \cdot * \cdot E \;|\;( \cdot E \cdot ) \;|\;N$ \\
+\end{tabular}}
+\end{center}
+
+Decide on how many precedence levels, say\medskip\\
+\hspace{5mm}highest for \bl{$()$}, medium for \bl{*}, lowest for \bl{+}
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$E_{low}$ & $\rightarrow$ & $E_{med} \cdot + \cdot E_{low} \;|\; E_{med}$ \\
+$E_{med}$ & $\rightarrow$ & $E_{hi} \cdot * \cdot E_{med} \;|\; E_{hi}$\\
+$E_{hi}$ & $\rightarrow$ &  $( \cdot E_{low} \cdot ) \;|\;N$ \\
+\end{tabular}}
+\end{center}\pause
+
+\small What happens with \bl{$1 + 3  + 4$}?
+\end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
 \begin{frame}[c]
-\frametitle{Palindromes}
+\frametitle{\begin{tabular}{c}Removing Left-Recursion\end{tabular}}
 
-A grammar for palindromes over the alphabet~\bl{$\{a,b\}$}:
+The rule for numbers is directly left-recursive:
 
 \begin{center}
 \bl{\begin{tabular}{lcl}
-$S$ & $\rightarrow$ &  $\epsilon$ \\
-$S$ & $\rightarrow$ &  $a\cdot S\cdot a$ \\
-$S$ & $\rightarrow$ &  $b\cdot S\cdot b$ \\
+$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1\;\;\;\;(\ldots)$ 
+\end{tabular}}
+\end{center}
+
+Translate
+
+\begin{center}
+\begin{tabular}{ccc}
+\bl{\begin{tabular}{lcl}
+$N$ & $\rightarrow$ & $N \cdot \alpha$\\
+ &  $\;|\;$ & $\beta$\\
+ \\ 
+\end{tabular}} 
+& {\Large$\Rightarrow$} &
+\bl{\begin{tabular}{lcl}
+$N$ & $\rightarrow$ & $\beta \cdot N'$\\
+$N'$ & $\rightarrow$ & $\alpha \cdot N'$\\
+ &  $\;|\;$ & $\epsilon$ 
 \end{tabular}}
+\end{tabular}
 \end{center}\pause
 
+Which means
+
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1 \cdot N'$\\
+$N'$ & $\rightarrow$ & $N \cdot N' \;|\; \epsilon$\\
+\end{tabular}}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Chomsky Normal Form\end{tabular}}
+
+All rules must be of the form
+
+\begin{center}
+\bl{$A \rightarrow a$}
+\end{center}
+
 or
 
 \begin{center}
-\bl{\begin{tabular}{lcl}
-$S$ & $\rightarrow$ &  $\epsilon \;|\; a\cdot S\cdot a \;|\;b\cdot S\cdot b$ \\
-\end{tabular}}
-\end{center}\pause\bigskip
-
-\small
-Can you find the grammar rules for matched parentheses?
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+\bl{$A \rightarrow B\cdot C$}
+\end{center}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Arithmetic Expressions}
+No rule can contain \bl{$\epsilon$}.
 
-\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
-
-\bl{\texttt{1 + 2 * 3 + 4}}
-
-\end{frame}
+\end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
 \begin{frame}[c]
-\frametitle{A CFG Derivation}
+\frametitle{\begin{tabular}{c}$\epsilon$-Removal\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
+\item If \bl{$A\rightarrow \alpha \cdot B \cdot \beta$} and \bl{$B \rightarrow \epsilon$} are in the grammar,
+then add \bl{$A\rightarrow \alpha \cdot \beta$} (iterate if necessary).
+\item Throw out all \bl{$B \rightarrow \epsilon$}.
 \end{enumerate}
 
-\begin{center}
-\bl{$S \rightarrow \ldots \rightarrow \ldots  \rightarrow \ldots  \rightarrow \ldots $}
-\end{center}
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-  
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Example Derivation}
-
+\small
 \begin{center}
-\bl{\begin{tabular}{lcl}
-$S$ & $\rightarrow$ &  $\epsilon \;|\; a\cdot S\cdot a \;|\;b\cdot S\cdot b$ \\
+\begin{tabular}{ccc}
+\bl{\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
+$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1\cdot N'$\\
+$N'$ & $\rightarrow$ & $N \cdot N'\;|\;\epsilon$\\
+\\ 
+\\
+\\
+\\
+\\
+\end{tabular}} &
+\bl{\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
+\\
+$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1\cdot N'\;|\;0\;|\;1$\\
+$N'$ & $\rightarrow$ & $N \cdot N'\;|\;N\;|\;\epsilon$\\
+\\
+$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1\cdot N'\;|\;0\;|\;1$\\
+$N'$ & $\rightarrow$ & $N \cdot N'\;|\;N$\\
 \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}
+\pause\normalsize
+\begin{center}
+\bl{\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
+$N$ & $\rightarrow$ & $0 \cdot N\;|\; 1\cdot N\;|\;0\;|\;1$\\
+\end{tabular}}
+
+\end{center}
+\end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
 \begin{frame}[c]
-\frametitle{Example Derivation}
+\frametitle{\begin{tabular}{c}CYK Algorithm\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
-
+If grammar is in Chomsky normalform \ldots
 
 \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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Language of a CFG}
-
-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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Parse Trees}
-
-\begin{center}
-\bl{\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $F \;|\; F \cdot * \cdot F$\\
-$F$ & $\rightarrow$ & $T \;|\; T \cdot + \cdot T \;|\; T \cdot - \cdot T$\\
-$T$ & $\rightarrow$ & $num\_token \;|\; ( \cdot E \cdot )$\\
+\bl{\begin{tabular}{@ {}lcl@ {}}
+$S$ & $\rightarrow$ &  $N\cdot P$ \\
+$P$ & $\rightarrow$ &  $V\cdot N$ \\
+$N$ & $\rightarrow$ &  $N\cdot N$ \\
+$N$ & $\rightarrow$ &  $\texttt{students} \;|\; \texttt{Jeff} \;|\; \texttt{geometry} \;|\; \texttt{trains} $ \\
+$V$ & $\rightarrow$ &  $\texttt{trains}$ 
 \end{tabular}}
 \end{center}
 
-\begin{center}
-\begin{tikzpicture}[level distance=8mm, blue]
-  \node {$E$}
-    child {node {$F$} 
-     child {node {$T$} 
-                 child {node {(\,$E$\,)}
-                            child {node{$F$ *{} $F$}
-                                  child {node {$T$} child {node {2}}}
-                                  child {node {$T$} child {node {3}}} 
-                               }
-                          }
-              }
-     child {node {+}}
-     child {node {$T$}
-       child {node {(\,$E$\,)} 
-       child {node {$F$}
-       child {node {$T$ +{} $T$}
-                    child {node {3}}
-                    child {node {4}} 
-                 }
-                 }}
-    }};
-\end{tikzpicture}
-\end{center}
+\bl{\texttt{Jeff trains geometry students}}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}CYK Algorithm\end{tabular}}
 
-\begin{textblock}{5}(1, 6.5)
-\bl{\texttt{(2*3)+(3+4)}}
-\end{textblock}
 
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+\begin{itemize}
+\item fastest possible algorithm for recognition problem
+\item runtime is \bl{$O(n^3)$}\bigskip
+\item grammars need to be transferred into CNF
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
 \begin{frame}[c]
-\frametitle{Arithmetic Expressions}
 
 \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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Ambiguous Grammars}
-
-A grammar is \alert{ambiguous} if there is a string that has
-at least two different parse trees.
-
-
-\begin{center}
-\bl{\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $num\_token$ \\
-$E$ & $\rightarrow$ &  $E \cdot + \cdot E$ \\
-$E$ & $\rightarrow$ &  $E \cdot - \cdot E$ \\
-$E$ & $\rightarrow$ &  $E \cdot * \cdot E$ \\
-$E$ & $\rightarrow$ &  $( \cdot E \cdot )$ 
+\bl{\begin{tabular}{@{}lcl@{}}
+\textit{Stmt} & $\rightarrow$ &  $\texttt{skip}$\\
+              & $|$ & \textit{Id}\;\texttt{:=}\;\textit{AExp}\\
+              & $|$ & \texttt{if}\; \textit{BExp} \;\texttt{then}\; \textit{Block} \;\texttt{else}\; \textit{Block}\\
+              & $|$ & \texttt{while}\; \textit{BExp} \;\texttt{do}\; \textit{Block}\\
+              & $|$ & \texttt{read}\;\textit{Id}\\
+              & $|$ & \texttt{write}\;\textit{Id}\\
+              & $|$ & \texttt{write}\;\textit{String}\medskip\\
+\textit{Stmts} & $\rightarrow$ &  \textit{Stmt} \;\texttt{;}\; \textit{Stmts}\\
+              & $|$ & \textit{Stmt}\medskip\\
+\textit{Block} & $\rightarrow$ &  \texttt{\{}\,\textit{Stmts}\,\texttt{\}}\\
+                & $|$ & \textit{Stmt}\medskip\\
+\textit{AExp} & $\rightarrow$ & \ldots\\
+\textit{BExp} & $\rightarrow$ & \ldots\\
 \end{tabular}}
 \end{center}
-
-\bl{\texttt{1 + 2 * 3 + 4}}
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Dangling Else}
-
-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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Parser Combinators}
-
-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 atomic parsers
-\item sequencing
-\item alternative
-\item semantic action
-\end{itemize}
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-
-Atomic parsers, for example, number tokens
-
-\begin{center}
-\bl{$\texttt{Num(123)}::rest \;\Rightarrow\; \{(\texttt{Num(123)}, rest)\}$} 
-\end{center}\bigskip
-
-\begin{itemize}
-\item you consume one or more token from the\\ 
-  input (stream)
-\item also works for characters and strings
-\end{itemize}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\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}
+\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:\medskip 
-\begin{center}
-((output$_1$, output$_2$), unparsed part)
-\end{center}
-\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}
-
+\mbox{\lstinputlisting[language=while]{../progs/fib.while}}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -449,227 +401,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Input Types of Parsers}
-
-\begin{itemize}
-\item input: \alert{token list}
-\item output: set of (output\_type, \alert{token list})
-\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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Scannerless Parsers}
-
-\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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Successful Parses}
-
-\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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Abstract Parser Class}
-
-\small
-\lstinputlisting[language=Scala,xleftmargin=1mm]{../progs/app7.scala}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-
-\small
-\fontsize{10}{12}\selectfont
-\lstinputlisting[language=Scala,xleftmargin=1mm]{../progs/app8.scala}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{Two Grammars}
-
-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}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[t]
-\frametitle{Ambiguous Grammars}
-
-\begin{center}
-\begin{tikzpicture}
-\begin{axis}[xlabel={\pcode{1}s},ylabel={time in secs},
-    enlargelimits=false,
-    xtick={0,100,...,1000},
-    xmax=1050,
-    ymax=33,
-    ytick={0,5,...,30},
-    scaled ticks=false,
-    axis lines=left,
-    width=11cm,
-    height=7cm, 
-    legend entries={unambiguous,ambiguous},  
-    legend pos=north east,
-    legend cell align=left,
-    x tick label style={font=\small,/pgf/number format/1000 sep={}}]
-\addplot[blue,mark=*, mark options={fill=white}] 
-  table {s-grammar1.data};
-\only<2>{
-  \addplot[red,mark=triangle*, mark options={fill=white}] 
-  table {s-grammar2.data};}  
-\end{axis}
-\end{tikzpicture}
-\end{center}
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-\begin{frame}
-\frametitle{While-Language}
-\mbox{}\\[-23mm]\mbox{}
-
-\bl{\begin{plstx}[rhs style=,one per line]
: \meta{Stmt} ::= skip
-         | \meta{Id} := \meta{AExp}
-         | if \meta{BExp} then \meta{Block} else \meta{Block}
-         | while \meta{BExp} do \meta{Block}\\
-: \meta{Stmts} ::= \meta{Stmt} ; \meta{Stmts}
-          | \meta{Stmt}\\
-: \meta{Block} ::= \{ \meta{Stmts} \}
-          | \meta{Stmt}\\
-: \meta{AExp} ::= \ldots\\
-: \meta{BExp} ::= \ldots\\
\end{plstx}}
-
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-\frametitle{An Interpreter}
+\frametitle{\begin{tabular}{c}An Interpreter\end{tabular}}
 
 \begin{center}
 \bl{\begin{tabular}{l}
@@ -684,12 +416,103 @@
 
 \begin{itemize}
 \item the interpreter has to record the value of \bl{$x$} before assigning a value to \bl{$y$}\pause
-\item \bl{\texttt{eval(stmt, env)}}
+\item \bl{\text{eval}(stmt, env)}
 \end{itemize}
 
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Interpreter\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{@{}lcl@{}}
+$\text{eval}(n, E)$ & $\dn$ & $n$\\
+$\text{eval}(x, E)$ & $\dn$ & $E(x)$ \;\;\;\textcolor{black}{lookup \bl{$x$} in \bl{$E$}}\\
+$\text{eval}(a_1 + a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) + \text{eval}(a_2, E)$\\
+$\text{eval}(a_1 - a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) - \text{eval}(a_2, E)$\\
+$\text{eval}(a_1 * a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) * \text{eval}(a_2, E)$\bigskip\\
+$\text{eval}(a_1 = a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) = \text{eval}(a_2, E)$\\
+$\text{eval}(a_1\,!\!= a_2, E)$ & $\dn$ & $\neg(\text{eval}(a_1, E) = \text{eval}(a_2, E))$\\
+$\text{eval}(a_1 < a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) < \text{eval}(a_2, E)$\
+\end{tabular}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Interpreter (2)\end{tabular}}
+
+\begin{center}
+\bl{\begin{tabular}{@{}lcl@{}}
+$\text{eval}(\text{skip}, E)$ & $\dn$ & $E$\\
+$\text{eval}(x:=a, E)$ & $\dn$ & \bl{$E(x \mapsto \text{eval}(a, E))$}\\
+\multicolumn{3}{@{}l@{}}{$\text{eval}(\text{if}\;b\;\text{then}\;cs_1\;\text{else}\;cs_2 , E) \dn$}\\
+\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{if}\;\text{eval}(b,E)\;\text{then}\;
+\text{eval}(cs_1,E)$}\\
+\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\phantom{\text{if}\;\text{eval}(b,E)\;}\text{else}\;\text{eval}(cs_2,E)$}\\
+\multicolumn{3}{@{}l@{}}{$\text{eval}(\text{while}\;b\;\text{do}\;cs, E) \dn$}\\
+\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{if}\;\text{eval}(b,E)$}\\
+\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{then}\;
+\text{eval}(\text{while}\;b\;\text{do}\;cs, \text{eval}(cs,E))$}\\
+\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{else}\; E$}\\
+$\text{eval}(\text{write}\; x, E)$ & $\dn$ & $\{\;\text{println}(E(x))\; ;\;E\;\}$\\
+\end{tabular}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Test Program\end{tabular}}
+
+\mbox{}\\[-18mm]\mbox{}
+
+{\lstset{language=While}%%\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting{../progs/loops.while}}}
+
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Interpreted Code\end{tabular}}
+
+\begin{center}
+\begin{tikzpicture}
+\begin{axis}[axis x line=bottom, axis y line=left, xlabel=n, ylabel=secs, legend style=small]
+\addplot+[smooth] file {interpreted.data};
+\end{axis}
+\end{tikzpicture}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Java Virtual Machine\end{tabular}}
+
+\begin{itemize}
+\item introduced in 1995
+\item is a stack-based VM (like Postscript, CLR of .Net)
+\item contains a JIT compiler
+\item many languages take advantage of JVM's infrastructure (JRE)
+\item is garbage collected $\Rightarrow$ no buffer overflows
+\item some languages compile to the JVM: Scala, Clojure\ldots
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
 
 \end{document}