\documentclass[dvipsnames,14pt,t]{beamer}
\usepackage{beamerthemeplaincu}
\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} 
\usepackage{../langs}
\usepackage{../data}


\makeatletter
\lst@CCPutMacro\lst@ProcessOther {"2D}{\lst@ttfamily{-{}}{-{}}}
\@empty\z@\@empty
\makeatother


% beamer stuff 
\renewcommand{\slidecaption}{AFL 06, King's College London, 30.~October 2013}
\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions

\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\\
  Office: & S1.27 (1st floor Strand Building)\\
  Slides: & KEATS (also home work is there)\\
  \end{tabular}
  \end{center}


\end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}

While regular expressions are very useful for lexing, 
there is no regular expression that can recognise the language \bl{$a^nb^n$}.\bigskip

\begin{center}
\bl{$(((()()))())$} \;\;vs.\;\; \bl{$(((()()))()))$}
\end{center}


\end{frame}}
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\newcommand{\qq}{\mbox{\texttt{"}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Grammars\end{tabular}}

A (context-free) grammar \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{$A \rightarrow \text{rhs}$}
\end{center}

where \bl{rhs} are sequences involving terminals and nonterminals,
including the empty sequence \bl{$\epsilon$}.\medskip\pause

We also allow rules
\begin{center}
\bl{$A \rightarrow \text{rhs}_1 | \text{rhs}_2 | \ldots$}
\end{center}
\end{itemize}


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Palindromes\end{tabular}}

\begin{center}
\bl{\begin{tabular}{lcl}
$S$ & $\rightarrow$ &  $\epsilon$ \\
$S$ & $\rightarrow$ &  $a\cdot S\cdot a$ \\
$S$ & $\rightarrow$ &  $b\cdot S\cdot b$ \\
\end{tabular}}
\end{center}\pause

or

\begin{center}
\bl{\begin{tabular}{lcl}
$S$ & $\rightarrow$ &  $\epsilon \;|\; a\cdot S\cdot a \;|\;b\cdot S\cdot b$ \\
\end{tabular}}
\end{center}


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   



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\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

\bl{\texttt{1 + 2 * 3 + 4}}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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]
\frametitle{\begin{tabular}{c}Parse Trees\end{tabular}}

\begin{center}
\bl{\begin{tabular}{lcl}
$E$ & $\rightarrow$ &  $F \;|\; F \cdot * \cdot F$\\
$F$ & $\rightarrow$ & $T \;|\; T \cdot + \cdot T \;|\; T \cdot - \cdot T$\\
$T$ & $\rightarrow$ & $num\_token \;|\; ( \cdot E \cdot )$\\
\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}

\begin{textblock}{5}(1, 6.5)
\bl{\texttt{(2*3)+(3+4)}}
\end{textblock}

\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 two different parse trees.


\begin{center}
\bl{\begin{tabular}{lcl}
$E$ & $\rightarrow$ &  $num\_token$ \\
$E$ & $\rightarrow$ &  $E \cdot + \cdot E$ \\
$E$ & $\rightarrow$ &  $E \cdot - \cdot E$ \\
$E$ & $\rightarrow$ &  $E \cdot * \cdot E$ \\
$E$ & $\rightarrow$ &  $( \cdot E \cdot )$ 
\end{tabular}}
\end{center}

\bl{\texttt{1 + 2 * 3 + 4}}

\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 Parser Class}

\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


  


\end{document}

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