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  morekeywords={abstract,case,catch,class,def,%
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	basicstyle=\ttfamily,
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% beamer stuff 
\renewcommand{\slidecaption}{AFL 08, King's College London, 21.~November 2012}
\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions


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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \\[-3mm]
  \LARGE Automata and \\[-2mm] 
  \LARGE Formal Languages (8)\\[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}Building a ``Web Browser''\end{tabular}}

Using a lexer: assume the following regular expressions

\begin{center}
\bl{\begin{tabular}{lcl}
$SY\!M$ & $\dn$ & $(\text{a}..\text{zA}..\text{Z0}..\text{9}..)$\\
$W\!O\!RD$ & $\dn$ & $SY\!M^+$\\
$BT\!AG$ & $\dn$ & $<\!W\!O\!RD\!>$\\
$ET\!AG$ & $\dn$ & $<\!/W\!O\!RD\!>$\\
$W\!HIT\!E$ & $\dn$ & $\texttt{" "} + \texttt{"}\slash{}n\texttt{"}$\\
\end{tabular}}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Interpreting a List of Tokens\end{tabular}}

\begin{itemize}
\item the text should be formatted consistently up to a specified width, say 60 characters 
\item potential linebreaks are inserted by the formatter (not the input)
\item repeated whitespaces are ``condensed'' to a single whitepace
\item \bl{$<\!p\!>$} \bl{$<\!\slash{}p\!>$}  start/end paragraph
\item \bl{$<\!b\!>$} \bl{$<\!\slash{}b\!>$}  start/end bold
\item \bl{$<\!red\!>$} \bl{$<\!\slash{}red\!>$}  start/end red (cyan, etc)


\end{itemize}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Interpreting a List of Tokens\end{tabular}}

The lexer cannot prevent errors like

\begin{center}
\bl{$<\!b\!>$} \ldots \bl{$<\!p\!>$} \ldots \bl{$<\!\slash{}b\!>$} \ldots \bl{$<\!\slash{}p\!>$} 
\end{center}

or

\begin{center}
\bl{$<\!\slash{}b\!>$} \ldots \bl{$<\!b\!>$}
\end{center}


\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{$p \Longrightarrow f$})\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]

Token parser:\bigskip

\begin{itemize}
\item if the input is

\begin{center}
\large \bl{$tok_1:: tok_2 :: \ldots :: tok_n$}
\end{center}

then return

\begin{center}
\large \bl{$\{(tok_1,\; tok_2 :: \ldots :: tok_n)\}$}
\end{center}

or

\begin{center}
\large \bl{$\{\}$}
\end{center}

if \bl{$tok_1$} is not the right token we are looking for
\end{itemize}



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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

Number-Token parser:\bigskip

\begin{itemize}
\item if the input is

\begin{center}
\large \bl{$num\_tok(42):: tok_2 :: \ldots :: tok_n$}
\end{center}

then return

\begin{center}
\large \bl{$\{(42,\; tok_2 :: \ldots :: tok_n)\}$}
\end{center}

or

\begin{center}
\large \bl{$\{\}$}
\end{center}

if \bl{$tok_1$} is not the right token we are looking for
\end{itemize}\pause

\begin{center}
list of tokens \bl{$\Rightarrow$} set of (\alert{int}, list of tokens)
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{itemize}
\item if the input is

\begin{center}
\begin{tabular}{l}
\large \bl{$num\_tok(42)::$}\\
\hspace{7mm}\large \bl{$num\_tok(3) ::$}\\
\hspace{14mm}\large \bl{$tok_3 :: \ldots :: tok_n$}
\end{tabular}
\end{center}

and the parser is 

\begin{center}
\bl{$ntp \sim ntp$}
\end{center}

the successful output will be

\begin{center}
\large \bl{$\{((42, 3),\; tok_2 :: \ldots :: tok_n)\}$}
\end{center}\pause

Now we can form
\begin{center}
\bl{$(ntp \sim ntp) \Longrightarrow f$}
\end{center}

where \bl{$f$} is the semantic action (``what to do with the pair'')

\end{itemize}

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

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

Addition

\begin{center}
\bl{$T \sim + \sim E \Longrightarrow \underbrace{f((x,y), z) \Rightarrow x + z}_{\text{semantic action}}$}
\end{center}\pause

Multiplication

\begin{center}
\bl{$F \sim * \sim T \Longrightarrow f((x,y), z) \Rightarrow x * z$}
\end{center}\pause

Parenthesis

\begin{center}
\bl{$\text{(} \sim E \sim \text{)} \Longrightarrow f((x,y), z) \Rightarrow y$}
\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{$p \Longrightarrow f$} 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{list of tokens}
\item output: set of (output\_type, \alert{list of tokens})
\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

\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]
{\lstset{language=Scala}\fontsize{10}{12}\selectfont
\texttt{\lstinputlisting{app7.scala}}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
{\lstset{language=Scala}\fontsize{10}{12}\selectfont
\texttt{\lstinputlisting{app7.scala}}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
{\lstset{language=Scala}\fontsize{10}{12}\selectfont
\texttt{\lstinputlisting{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}What about Left-Recursion?\end{tabular}}

\begin{itemize}
\item we record when we recursively called a parser\bigskip
\item whenever the is a recursion, the parser must have consumed something --- so
we can decrease the input string/list of token by one (at the end)
\end{itemize}
\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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