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

\hfuzz=220pt 

\pgfplotsset{compat=1.11}

\newcommand{\bl}[1]{\textcolor{blue}{#1}}  

\renewcommand{\slidecaption}{CFL 04, King's College London}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \\[-3mm]
  \LARGE Compilers and \\[-2mm] 
  \LARGE Formal Languages (4)\\[3mm] 
  \end{tabular}}

  \normalsize
  \begin{center}
  \begin{tabular}{ll}
  Email:  & christian.urban at kcl.ac.uk\\
  Office: & N\liningnums{7.07} (North Wing, Bush House)\\
  Slides: & KEATS (also homework is there)\\
  \end{tabular}
  \end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Survey: Thanks!}
\small

\hspace{9mm}{\it``\ldots Thanks a million! Thanks without end!''}

\begin{textblock}{3}(12,1.7)
\includegraphics[scale=0.1]{../pics/stickman.jpg}
\end{textblock}\pause

\begin{textblock}{6}(1,4.15)
\begin{bubble}[6.7cm]
  \it ``Urban is a very talented lecturer: thorough, concise,
  clear, and cares to make sure that we are learning!''
\end{bubble}
\end{textblock}

\only<3>{
\begin{textblock}{6}(0.6,8.5)
\includegraphics[scale=0.2]{../pics/s1.png}
\end{textblock}
\begin{textblock}{6}(8,8.5)
\includegraphics[scale=0.2]{../pics/s2.png}
\end{textblock}}

\only<4->{
\begin{textblock}{6}(0.6,8.5)
\includegraphics[scale=0.2]{../pics/s3.png}
\end{textblock}
\begin{textblock}{6}(8,8.5)
\includegraphics[scale=0.2]{../pics/s4.png}
\end{textblock}}

\only<5>{
\begin{textblock}{6}(1.5,14.5)  
  room too hot, 3h lecture
\end{textblock}}  

% \begin{itemize}
% \item {\bf My Voice} ``lecturer speaks in a low voice and 
%   is hard to hear him'' ``please use mic'' ``please use mic 
%   \& lecture recording''
% \item {\bf Pace} ``faster pace'' ``a bit quick for me 
% personally''
% \item {\bf Recording} ``please use recording class''
% \item {\bf Module Name} ``misleading''
% \item {\bf Examples} ``more examples''
% \item {\bf Assessment} ``really appreciate extension of 
%   first coursework'' 
% \end{itemize}
  
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

\begin{center}
\begin{tikzpicture}[scale=2,>=stealth',very thick,
                             every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
  \only<-7>{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
  \only<8->{\node[state, initial,accepting] (q0) at ( 0,1) {$\mbox{Q}_0$};}                           
  \only<-7>{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
  \only<8->{\node[state,accepting] (q1) at ( 1,1) {$\mbox{Q}_1$};}
  \node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};
  \path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
                  (q1) edge[bend left] node[above] {\alert{$b$}} (q0)
                  (q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
                  (q1) edge node[above] {\alert{$a$}} (q2)
                  (q2) edge [loop right] node {\alert{$a$}} ()
                  (q0) edge [loop below] node {\alert{$b$}} ()
            ;
\end{tikzpicture}
\end{center}\bigskip

\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,b + \mbox{Q}_1\,b +  \mbox{Q}_2\,b$}\\
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a$}\\
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_1\,a + \mbox{Q}_2\,a$}\\
\end{tabular}
\end{center}


Arden's Lemma:
\begin{center}
If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
\end{center}

\only<2-6>{\small
\begin{textblock}{6}(1,0.8)
\begin{bubble}[6.7cm]
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
\multicolumn{3}{@{}l}{substitute \bl{$\mbox{Q}_1$} into \bl{$\mbox{Q}_0$} \& \bl{$\mbox{Q}_2$}:}\\    
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,b + \mbox{Q}_0\,a\,b +  \mbox{Q}_2\,b$}\\
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
\end{tabular}
\end{bubble}
\end{textblock}}

\only<3-6>{\small
\begin{textblock}{6}(2,4.15)
\begin{bubble}[6.7cm]
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
\multicolumn{3}{@{}l}{simplifying \bl{$\mbox{Q}_0$}:}\\    
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b$}\\
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
\end{tabular}
\end{bubble}
\end{textblock}}

\only<4-6>{\small
\begin{textblock}{6}(3,7.55)
\begin{bubble}[6.7cm]
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  \multicolumn{3}{@{}l}{Arden for \bl{$\mbox{Q}_2$}:}\\    
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b$}\\
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a\,(a^*)$}
\end{tabular}
\end{bubble}
\end{textblock}}

\only<5-6>{\small
\begin{textblock}{6}(4,10.9)
\begin{bubble}[7.5cm]
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  \multicolumn{3}{@{}l}{Substitute \bl{$\mbox{Q}_2$} and simplify:}\\    
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\ONE + \mbox{Q}_0\,(b + a\,b + a\,a\,(a^*)\,b)$}\\
\end{tabular}
\end{bubble}
\end{textblock}}

\only<6>{\small
\begin{textblock}{6}(5,13.4)
\begin{bubble}[7.5cm]
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  \multicolumn{3}{@{}l}{Arden again for \bl{$\mbox{Q}_0$}:}\\    
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\
\end{tabular}
\end{bubble}
\end{textblock}}


\only<7->{\small
\begin{textblock}{6}(6,11.5)
\begin{bubble}[6.7cm]
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
\multicolumn{3}{@{}l}{Finally:}\\    
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a$}\\
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a\,a\,(a^*)$}\\
\end{tabular}
\end{bubble}
\end{textblock}}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Regexps and Automata}

\begin{center}
\begin{tikzpicture}
\node (rexp)  {\bl{\bf Regexps}};
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};
\path[->,red, line width=2mm] (rexp) edge node [above=4mm, black] 
     {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
\path[->,red, line width=2mm] (nfa) edge node [above=4mm, black] 
     {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
\path[->, red, line width=2mm] (dfa) edge node [below=5mm, black] {minimisation} (mdfa);
%%\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);
\path[->, red, line width=2mm] (dfa) edge [bend left=45] node [below, black] {\begin{tabular}{l}Brzozowski's\\ method\end{tabular}} (rexp);

\end{tikzpicture}\\
\end{center}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\bl{$a^{?\{n\}} \cdot a^{\{n\}}$}}

\begin{tikzpicture}
\begin{axis}[xlabel={\pcode{a}s},ylabel={time in secs},
    enlargelimits=false,
    xtick={0,5,...,30},
    xmax=30,
    ymax=35,
    ytick={0,5,...,30},
    scaled ticks=false,
    axis lines=left,
    width=10cm,
    height=7cm, 
    legend entries={Python,Ruby,my NFA},  
    legend pos=north west,
    legend cell align=left]
\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};  
\addplot[red,mark=triangle*, mark options={fill=white}] table {nfasearch.data};	 
\end{axis}
\end{tikzpicture}

The punchline is that many existing libraries do depth-first search
in NFAs (backtracking).

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

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[c]
% \frametitle{DFA to Rexp}

% \begin{center}
% \begin{tikzpicture}[scale=2,>=stealth',very thick,
%                     every state/.style={minimum size=0pt,
%                     draw=blue!50,very thick,fill=blue!20},]
%   \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[bend left] node[above] {\alert{$a$}} (q1)
%             (q1) edge[bend left] node[above] {\alert{$b$}} (q0)
%             (q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
%             (q1) edge node[above] {\alert{$a$}} (q2)
%             (q2) edge [loop right] node {\alert{$a$}} ()
%             (q0) edge [loop below] node {\alert{$b$}} ();
% \end{tikzpicture}
% \end{center}\bigskip

% \begin{center}
% \begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l@{\hspace{7mm}}l}
% \bl{$q_0$} & \bl{$=$} & \bl{$\ONE + q_0\,b + q_1\,b +  q_2\,b$} & (start state)\\
% \bl{$q_1$} & \bl{$=$} & \bl{$q_0\,a$}\\
% \bl{$q_2$} & \bl{$=$} & \bl{$q_1\,a + q_2\,a$}\\

% \end{tabular}
% \end{center}

% Arden's Lemma:
% \begin{center}
% If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
% \end{center}


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

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[c]
% \frametitle{DFA Minimisation}

% \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{$c$} test whether
% \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 no change.
% \item All unmarked pairs can be merged.
% \end{enumerate}

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

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[c]

% \begin{center}
% \begin{tabular}{@{\hspace{-8mm}}cc@{}}
% \begin{tikzpicture}[>=stealth',very thick,auto,
%                              every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
% \node[state,initial]  (q_0)  {$q_0$};
% \node[state] (q_1) [right=of q_0] {$q_1$};
% \node[state] (q_2) [below right=of q_0] {$q_2$};
% \node[state] (q_3) [right=of q_2] {$q_3$};
% \node[state, accepting] (q_4) [right=of q_1] {$q_4$};
% \path[->] (q_0) edge node [above]  {\alert{$a$}} (q_1);
% \path[->] (q_1) edge node [above]  {\alert{$a$}} (q_4);
% \path[->] (q_4) edge [loop right] node  {\alert{$a, b$}} ();
% \path[->] (q_3) edge node [right]  {\alert{$a$}} (q_4);
% \path[->] (q_2) edge node [above]  {\alert{$a$}} (q_3);
% \path[->] (q_1) edge node [right]  {\alert{$b$}} (q_2);
% \path[->] (q_0) edge node [above]  {\alert{$b$}} (q_2);
% \path[->] (q_2) edge [loop left] node  {\alert{$b$}} ();
% \path[->] (q_3) edge [bend left=95, looseness=1.3] node [below]  {\alert{$b$}} (q_0);
% \end{tikzpicture}
% &
% \raisebox{9mm}{\begin{tikzpicture}[scale=0.6,line width=0.8mm]
% \draw (0,0) -- (4,0);
% \draw (0,1) -- (4,1);
% \draw (0,2) -- (3,2);
% \draw (0,3) -- (2,3);
% \draw (0,4) -- (1,4);

% \draw (0,0) -- (0, 4);
% \draw (1,0) -- (1, 4);
% \draw (2,0) -- (2, 3);
% \draw (3,0) -- (3, 2);
% \draw (4,0) -- (4, 1);

% \draw (0.5,-0.5) node {$q_0$}; 
% \draw (1.5,-0.5) node {$q_1$}; 
% \draw (2.5,-0.5) node {$q_2$}; 
% \draw (3.5,-0.5) node {$q_3$};
 
% \draw (-0.5, 3.5) node {$q_1$}; 
% \draw (-0.5, 2.5) node {$q_2$}; 
% \draw (-0.5, 1.5) node {$q_3$}; 
% \draw (-0.5, 0.5) node {$q_4$}; 

% \draw (0.5,0.5) node {\large$\star$}; 
% \draw (1.5,0.5) node {\large$\star$}; 
% \draw (2.5,0.5) node {\large$\star$}; 
% \draw (3.5,0.5) node {\large$\star$};
% \draw (0.5,1.5) node {\large$\star$}; 
% \draw (2.5,1.5) node {\large$\star$}; 
% \draw (0.5,3.5) node {\large$\star$}; 
% \draw (1.5,2.5) node {\large$\star$}; 
% \end{tikzpicture}}
% \end{tabular}
% \end{center}


% \mbox{}\\[-20mm]\mbox{}

% \begin{center}
% \begin{tikzpicture}[>=stealth',very thick,auto,
%                              every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
% \node[state,initial]  (q_02)  {$q_{0, 2}$};
% \node[state] (q_13) [right=of q_02] {$q_{1, 3}$};
% \node[state, accepting] (q_4) [right=of q_13] {$q_{4\phantom{,0}}$};
% \path[->] (q_02) edge [bend left] node [above]  {\alert{$a$}} (q_13);
% \path[->] (q_13) edge [bend left] node [below]  {\alert{$b$}} (q_02);
% \path[->] (q_02) edge [loop below] node  {\alert{$b$}} ();
% \path[->] (q_13) edge node [above]  {\alert{$a$}} (q_4);
% \path[->] (q_4) edge [loop above] node  {\alert{$a, b$}} ();
% \end{tikzpicture}\\
% minimal automaton
% \end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Regular Languages}

Two equivalent definitions:\bigskip

\begin{quote}\rm A language is \alert{regular} iff there exists a
regular expression that recognises all its strings.
\end{quote}

\begin{quote}\rm A language is \alert{regular} iff there exists an
automaton that recognises all its strings.
\end{quote}\bigskip\bigskip


\small
for example \bl{$a^nb^n$} is not regular
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Negation}

Regular languages are closed under negation:\bigskip

\begin{center}
\begin{tikzpicture}[scale=2,>=stealth',very thick,
                    every state/.style={minimum size=0pt,
                    draw=blue!50,very thick,fill=blue!20}]
  \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] {\alert{$a$}} (q1)
            (q1) edge[bend left] node[above] {\alert{$b$}} (q0)
            (q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
            (q1) edge node[above] {\alert{$a$}} (q2)
            (q2) edge [loop right] node {\alert{$a$}} ()
            (q0) edge [loop below] node {\alert{$b$}} ();
\end{tikzpicture}
\end{center}\bigskip\bigskip

But requires that the automaton is \alert{completed}!

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Goal of this Course}
\mbox{}\\[-26mm]\mbox{}

\begin{center}
  \begin{tikzpicture}[scale=1,
                      node/.style={
                      rectangle,rounded corners=3mm,
                      very thick,draw=black!50,
                      minimum height=18mm, minimum width=20mm,
                      top color=white,bottom color=black!20}]

  \node at (3.05, 1.8) {\Large\bf Write a compiler};

  \node (0) at (-2.3,0) {}; 
  
  \node (A) at (0,0)  [node] {};
  \node [below right] at (A.north west) {lexer};

  \node (B) at (3,0)  [node] {};
  \node [below right=1mm] at (B.north west) 
    {\mbox{}\hspace{-1mm}parser};

  \node (C) at (6,0)  [node] {};
  \node [below right] at (C.north west) 
    {\mbox{}\hspace{-1mm}code gen};

  \node (1) at (8.4,0) {}; 

  \draw [->,line width=4mm] (0) -- (A); 
  \draw [->,line width=4mm] (A) -- (B); 
  \draw [->,line width=4mm] (B) -- (C); 
  \draw [->,line width=4mm] (C) -- (1); 
  \end{tikzpicture}
  \end{center}
  
Today a lexer.  

\only<2>{
\begin{textblock}{1}(6,9.8)
\begin{tabular}{c}
\includegraphics[scale=0.13]{../pics/rosetta.jpg}\\[-2mm]
\footnotesize lexing $\Rightarrow$ recognising words (Stone of Rosetta)
\end{tabular}
\end{textblock}}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Regular Expressions}

In programming languages they are often used to recognise:\medskip

\begin{itemize}
\item symbols, digits
\item identifiers
\item numbers (non-leading zeros)
\item keywords
\item comments
\end{itemize}\bigskip

\mbox{}\hfill\bl{\url{http://www.regexper.com}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Lexing: Test Case}

\mbox{\lstinputlisting[language=While]{../progs/fib.while}}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\begin{frame}[c]
%\frametitle{?? Test Case}
%
%\mbox{\lstinputlisting[language=While]{../progs/collatz.while}}
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]

\tt
\begin{center}\large
\code{"if true then then 42 else +"}
\end{center}


\begin{tabular}{@{}l}
KEYWORD: \\
\hspace{5mm}{if}, {then}, {else},\\ 
WHITESPACE:\\
\hspace{5mm}{" "}, {$\backslash$n},\\ 
IDENTIFIER:\\
\hspace{5mm}LETTER $\cdot$ (LETTER + DIGIT + {\_})$^*$\\ 
NUM:\\
\hspace{5mm}(NONZERODIGIT $\cdot$ DIGIT$^*$) + {0}\\
OP:\\
\hspace{5mm}+, -, *, \%, <, <=\\
COMMENT:\\
\hspace{5mm}{$\slash$*} $\cdot$ $\sim$(ALL$^*$ $\cdot$ (*$\slash$) $\cdot$ ALL$^*$) $\cdot$ {*$\slash$}
\end{tabular}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]

\tt
\begin{center}\large
\code{"if true then then 42 else +"}
\end{center}

\only<1>{
\small\begin{tabular}{l}
KEYWORD(if),\\ 
WHITESPACE,\\ 
IDENT(true),\\ 
WHITESPACE,\\ 
KEYWORD(then),\\ 
WHITESPACE,\\ 
KEYWORD(then),\\ 
WHITESPACE,\\ 
NUM(42),\\ 
WHITESPACE,\\ 
KEYWORD(else),\\ 
WHITESPACE,\\ 
OP(+)
\end{tabular}}

\only<2>{
\small\begin{tabular}{l}
KEYWORD(if),\\ 
IDENT(true),\\ 
KEYWORD(then),\\ 
KEYWORD(then),\\ 
NUM(42),\\ 
KEYWORD(else),\\ 
OP(+)
\end{tabular}}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]


There is one small problem with the tokenizer. How should we 
tokenize\ldots?

\begin{center}
\large\code{"x-3"}
\end{center}

\tt
\begin{tabular}{@{}l}
ID: \ldots\\
OP:\\
\hspace{5mm}\texttt{"+"}, \texttt{"-"}\\
NUM:\\
\hspace{5mm}(NONZERODIGIT $\cdot$ DIGIT$^*$) + {''0''}\\
NUMBER:\\
\hspace{5mm}NUM +  (\texttt{"-"} $\cdot$ NUM)\\
\end{tabular}

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

\begin{frame}[c]

The same problem with\medskip

\[
\bl{(ab + a) \cdot (c + bc)}
\]\medskip

and the string $\bl{abc}$.\pause\pause\bigskip

Or, keywords are \pcode{if} and identifiers are 
letters followed by ``letters + numbers + \_''$^*$

\[
\bl{if}\qquad\bl{iffoo}
\]

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{POSIX: Two Rules}

\begin{itemize}
\item Longest match rule (``maximal munch rule''): The 
longest initial substring matched by any regular expression is taken
as the next token.\bigskip

\item Rule priority:
For a particular longest initial substring, the first regular
expression that can match determines the token.
\end{itemize}\bigskip\bigskip\pause

\small
\hfill most posix matchers are buggy\\
\footnotesize
\hfill 
\url{http://www.haskell.org/haskellwiki/Regex_Posix}
\smallskip\pause

\hfill \small traditional lexers are fast, but hairy

%\url{http://www.technologyreview.com/tr10/?year=2011}  
%finite deterministic automata/ nondeterministic automaton
%\item problem with infix operations, for example i-12

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Sulzmann \& Lu Matcher}

We want to match the string \bl{$abc$} using \bl{$r_1$}:

\begin{center}
\begin{tikzpicture}[scale=2,node distance=1.3cm,every node/.style={minimum size=8mm}]
\node (r1)  {\bl{$r_1$}};
\node (r2) [right=of r1] {\bl{$r_2$}};
\draw[->,line width=1mm]  (r1) -- (r2) node[above,midway] {\bl{$der\,a$}};\pause
\node (r3) [right=of r2] {\bl{$r_3$}};
\draw[->,line width=1mm]  (r2) -- (r3) node[above,midway] {\bl{$der\,b$}};\pause
\node (r4) [right=of r3] {\bl{$r_4$}};
\draw[->,line width=1mm]  (r3) -- (r4) node[above,midway] {\bl{$der\,c$}};\pause
\draw (r4) node[anchor=west] {\;\raisebox{3mm}{\bl{$nullable?$}}};\pause
\node (v4) [below=of r4] {\bl{$v_4$}};
\draw[->,line width=1mm]  (r4) -- (v4);\pause
\node (v3) [left=of v4] {\bl{$v_3$}};
\draw[->,line width=1mm]  (v4) -- (v3) node[below,midway] {\bl{$inj\,c$}};\pause
\node (v2) [left=of v3] {\bl{$v_2$}};
\draw[->,line width=1mm]  (v3) -- (v2) node[below,midway] {\bl{$inj\,b$}};\pause
\node (v1) [left=of v2] {\bl{$v_1$}};
\draw[->,line width=1mm]  (v2) -- (v1) node[below,midway] {\bl{$inj\,a$}};\pause
\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{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Regexes and Values}

Regular expressions and their corresponding values:

\begin{center}
\begin{columns}
\begin{column}{3cm}
\begin{tabular}{@{}rrl@{}}
  \bl{$r$} & \bl{$::=$}  & \bl{$\ZERO$}\\
           & \bl{$\mid$} & \bl{$\ONE$}   \\
           & \bl{$\mid$} & \bl{$c$}          \\
           & \bl{$\mid$} & \bl{$r_1 \cdot r_2$}\\
           & \bl{$\mid$} & \bl{$r_1 + r_2$}   \\
  \\
           & \bl{$\mid$} & \bl{$r^*$}         \\
  \\
  \end{tabular}
\end{column}
\begin{column}{3cm}
\begin{tabular}{@{\hspace{-7mm}}rrl@{}}
  \bl{$v$} & \bl{$::=$}  & \\
           &             & \bl{$Empty$}   \\
           & \bl{$\mid$} & \bl{$Char(c)$}          \\
           & \bl{$\mid$} & \bl{$Seq(v_1,v_2)$}\\
           & \bl{$\mid$} & \bl{$Left(v)$}   \\
           & \bl{$\mid$} & \bl{$Right(v)$}  \\
           & \bl{$\mid$} & \bl{$Stars\,[]$}      \\
           & \bl{$\mid$} & \bl{$Stars\,[v_1,\ldots\,v_n]$} \\
  \end{tabular}
\end{column}
\end{columns}
\end{center}

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

\begin{frame}[c]
\small

{\small\lstinputlisting[language=Scala,numbers=none,
xleftmargin=-5mm] {../progs/app01.scala}}

{\small\lstinputlisting[language=Scala,numbers=none,
xleftmargin=-5mm] {../progs/app02.scala}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Mkeps}

Finding a (posix) value for recognising the empty string:

\begin{center}
\begin{tabular}{lcl}
  \bl{$mkeps\,(\ONE)$}  & \bl{$\dn$}  & \bl{$Empty$}\\
  \bl{$mkeps\,(r_1 + r_2)$} & \bl{$\dn$}  & \bl{if $nullable(r_1)$}  \\
                          &             & \bl{then $Left(mkeps(r_1))$}\\
                          &             & \bl{else $Right(mkeps(r_2))$}\\
  \bl{$mkeps\,(r_1 \cdot r_2)$}  & \bl{$\dn$} & \bl{$Seq(mkeps(r_1),mkeps(r_2))$}\\
  \bl{$mkeps\,(r^*)$}      & \bl{$\dn$} & \bl{$Stars\,[]$}  \\
\end{tabular}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Inject}

Injecting (``Adding'') a character to a value\\[-12mm]\mbox{}

\begin{center}
\begin{tabular}{@{\hspace{-3mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
  \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{$inj\,(r^*)\,c\,(Seq(v,Stars\,vs))$} & \bl{$\dn$}  & \bl{$Stars\,(inj\,r\,c\,v\,::\,vs)$}\\
\end{tabular}
\end{center}\bigskip

\footnotesize
\bl{$inj$}: 1st arg $\mapsto$ a rexp; 2nd arg $\mapsto$ a character; 3rd arg $\mapsto$ a value 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

\begin{textblock}{10}(3,5)
\begin{tikzpicture}[scale=2,node distance=1.3cm,every node/.style={minimum size=8mm}]
\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}

\only<2->{
\begin{textblock}{6}(1,0.8)
\begin{bubble}[6cm]
\small
\begin{tabular}{ll}
\bl{$r_1$}: & \bl{$a \cdot (b \cdot c)$}\\
\bl{$r_2$}: & \bl{$\ONE \cdot (b \cdot c)$}\\
\bl{$r_3$}: & \bl{$(\ZERO \cdot (b \cdot c)) + (\ONE \cdot c)$}\\
\bl{$r_4$}: & \bl{$(\ZERO \cdot (b \cdot c)) + ((\ZERO \cdot c) + \ONE)$}\\
\end{tabular}
\end{bubble}
\end{textblock}}

\only<2->{
\begin{textblock}{6}(5,11.4)
\begin{bubble}[7.6cm]
\small
\begin{tabular}{ll}
\bl{$v_1$}: & \bl{$Seq(Char(a), Seq(Char(b), Char(c)))$}\\
\bl{$v_2$}: & \bl{$Seq(Empty, Seq(Char(b), Char(c)))$}\\
\bl{$v_3$}: & \bl{$Right(Seq(Empty, Char(c)))$}\\
\bl{$v_4$}: & \bl{$Right(Right(Empty))$}\\
\end{tabular}
\end{bubble}
\end{textblock}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Flatten}

Obtaining the string underlying a value:

\begin{center}
\begin{tabular}{lcl}
  \bl{$|Empty|$}     & \bl{$\dn$} & \bl{$[]$}\\
  \bl{$|Char(c)|$}   & \bl{$\dn$} & \bl{$[c]$}\\
  \bl{$|Left(v)|$}   & \bl{$\dn$} & \bl{$|v|$}\\
  \bl{$|Right(v)|$}  & \bl{$\dn$} & \bl{$|v|$}\\
  \bl{$|Seq(v_1,v_2)|$}& \bl{$\dn$} & \bl{$|v_1| \,@\, |v_2|$}\\
  \bl{$|[v_1,\ldots ,v_n]|$} & \bl{$\dn$} & \bl{$|v_1| \,@\ldots @\, |v_n|$}\\
\end{tabular}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

\begin{textblock}{10}(3,5)
\begin{tikzpicture}[scale=2,node distance=1.3cm,every node/.style={minimum size=8mm}]
\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{textblock}{6}(1,0.8)
\begin{bubble}[6cm]
\small
\begin{tabular}{ll}
\bl{$r_1$}: & \bl{$a \cdot (b \cdot c)$}\\
\bl{$r_2$}: & \bl{$\ONE \cdot (b \cdot c)$}\\
\bl{$r_3$}: & \bl{$(\ZERO \cdot (b \cdot c)) + (\ONE \cdot c)$}\\
\bl{$r_4$}: & \bl{$(\ZERO \cdot (b \cdot c)) + ((\ZERO \cdot c) + \ONE)$}\\
\end{tabular}
\end{bubble}
\end{textblock}

\begin{textblock}{6}(1,11.4)
\begin{bubble}[7.6cm]
\small
\begin{tabular}{ll}
\bl{$v_1$}: & \bl{$Seq(Char(a), Seq(Char(b), Char(c)))$}\\
\bl{$v_2$}: & \bl{$Seq(Empty, Seq(Char(b), Char(c)))$}\\
\bl{$v_3$}: & \bl{$Right(Seq(Empty, Char(c)))$}\\
\bl{$v_4$}: & \bl{$Right(Right(Empty))$}\\
\end{tabular}
\end{bubble}
\end{textblock}

\begin{textblock}{6}(12,11.4)
\begin{bubble}[2cm]
\small
\begin{tabular}{ll}
\bl{$|v_1|$}: & \bl{$abc$}\\
\bl{$|v_2|$}: & \bl{$bc$}\\
\bl{$|v_3|$}: & \bl{$c$}\\
\bl{$|v_4|$}: & \bl{$[]$}
\end{tabular}
\end{bubble}
\end{textblock}


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Lexing}

\begin{center}
\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
  \bl{$lex\,r\,[]$} & \bl{$\dn$}  & \bl{if $nullable(r)$ then $mkeps(r)$ else $error$}\\
  \bl{$lex\,r\,c::s$} & \bl{$\dn$}  & \bl{$inj\,r\,c\,lex(der(c,r), s)$}\\
\end{tabular}
\end{center}

\footnotesize
\bl{$lex$}: returns a value

\begin{center}
\begin{tikzpicture}[scale=2,node distance=1.0cm,every node/.style={minimum size=6mm}]
\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{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Records}

\begin{itemize}
\item new regex: \bl{$(x:r)$}\hspace{7mm}new value: \bl{$Rec(x,v)$}\medskip\pause

\item \bl{$nullable(x:r) \dn nullable(r)$}
\item \bl{$der\,c\,(x:r) \dn (x:der\,c\,r)$}
\item \bl{$mkeps(x:r) \dn Rec(x, mkeps(r))$}
\item \bl{$inj\,(x:r)\,c\,v \dn Rec(x, inj\,r\,c\,v)$}
\end{itemize}\bigskip\bigskip\pause

\small
for extracting subpatterns \bl{$(z: ((x:ab) + (y:ba))$}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

\begin{itemize}
\item A regular expression for email addresses

\begin{center}
\begin{tabular}{l}
(name: \bl{$[a\mbox{-}z0\mbox{-}9\_\!\_\,.-]^+$})\bl{$\cdot @\cdot$}\\ 
\qquad(domain: \bl{$[a\mbox{-}z0\mbox{-}9\,.-]^+$}) \bl{$\cdot .\cdot$}\\ 
\qquad\qquad(top\_level: \bl{$[a\mbox{-}z\,.]^{\{2,6\}}$})
\end{tabular}
\end{center}

\bl{\[
\texttt{christian.urban@kcl.ac.uk}
\]}

\item the result environment:

\begin{center}
\begin{tabular}{l}
\bl{$[(name:\texttt{christian.urban}),$}\\ 
\bl{$\phantom{[}(domain:\texttt{kcl}),$}\\ 
\bl{$\phantom{[}(top\_level:\texttt{ac.uk})]$}
\end{tabular}
\end{center}
\end{itemize}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{While Tokens}

\begin{center}
\begin{tabular}{rcl}
\pcode{WHILE\_REGS} & $\dn$ & \raisebox{-1mm}{\large(}\pcode{("k" : KEYWORD)} +\\ 
                  &       & \phantom{(}\pcode{("i" : ID)} +\\ 
                  &       & \phantom{(}\pcode{("o" : OP)} + \\
                  &       & \phantom{(}\pcode{("n" : NUM)} + \\
                  &       & \phantom{(}\pcode{("s" : SEMI)} +\\ 
                  &       & \phantom{(}\pcode{("p" : (LPAREN + RPAREN))} +\\ 
                  &       & \phantom{(}\pcode{("b" : (BEGIN + END))} +\\ 
                  &       & \phantom{(}\pcode{("w" : WHITESPACE)}\raisebox{-1mm}{\large)$^*$}
\end{tabular}
\end{center}

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



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Simplification}

\begin{itemize}
\item If we simplify after the derivative, then we are building the
value for the simplified regular expression, but \alert{\textbf{not}} for the original
regular expression.
\end{itemize}

\begin{center}
\begin{tikzpicture}[scale=2,node distance=1.3cm,every node/.style={minimum size=8mm}]
\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{center}

\small
\hspace{4.5cm}\bl{$(\ZERO \cdot (b \cdot c)) + ((\ZERO \cdot c) + \ONE)$}
$\mapsto$
\bl{$\ONE$}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

  Normally we would have

  \begin{center}
  \bl{$(\ZERO \cdot (b \cdot c)) + ((\ZERO \cdot c) + \ONE)$}
  \end{center}

  and answer how this regular expression matches the empty string
  with the value

  \begin{center}
  \bl{$Right(Right(Empty))$}
  \end{center}\bigskip

  But now we simplify this to \bl{$\ONE$} and would produce \bl{$Empty$} (see \textit{mkeps}).

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Rectification}

\def\arraystretch{1.05}
\begin{center}
\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l@{\hspace{8mm}}l}
& & & \hspace{5mm}rectification \\
& & & \hspace{5mm}functions:\\
\bl{$r \cdot \ZERO$} & $\mapsto$ & \bl{$\ZERO$} & \\ 
\bl{$\ZERO \cdot r$} & $\mapsto$ & \bl{$\ZERO$} & \\ 
\bl{$r \cdot \ONE$} & $\mapsto$ & \bl{$r$} & \bl{$\lambda f_1\,f_2\,v.\, Seq(f_1\,v, f_2\,Empty)$}\\ 
\bl{$\ONE \cdot r$} & $\mapsto$ & \bl{$r$} & \bl{$\lambda f_1\,f_2\,v.\, Seq(f_1\,Empty, f_2\,v)$}\\ 
\bl{$r + \ZERO$} & $\mapsto$ & \bl{$r$}   & \bl{$\lambda f_1\,f_2\,v.\, Left(f_1\,v)$}\\ 
\bl{$\ZERO + r$} & $\mapsto$ & \bl{$r$}   & \bl{$\lambda f_1\,f_2\,v.\, Right(f_2\,v)$}\\
\bl{$r + r$} & $\mapsto$ & \bl{$r$} & \bl{$\lambda f_1\,f_2\,v.\, Left(f_1\,v)$}
\end{tabular}
\end{center}\medskip\pause

\small
old \bl{$simp$} returns a rexp;\\
new \bl{$simp$} returns a rexp and a rectification~function.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Rectification}

\begin{center}
\begin{tabular}{l}
\bl{$simp(r)$}:\\
\quad case \bl{$r = r_1 + r_2$}\\
\qquad let \bl{$(r_{1s}, f_{1s}) = simp(r_1)$}\\
\qquad \phantom{let} \bl{$(r_{2s}, f_{2s}) = simp(r_2)$}\smallskip\\
\qquad case \bl{$r_{1s} = \ZERO$}: 
       return \bl{$(r_{2s}, \lambda v. \,Right(f_{2s}(v)))$}\\
\qquad case \bl{$r_{2s} = \ZERO$}: 
       return \bl{$(r_{1s}, \lambda v. \,Left(f_{1s}(v)))$}\\
\qquad case \bl{$r_{1s} = r_{2s}$}:
       return \bl{$(r_{1s}, \lambda v. \,Left(f_{1s}(v)))$}\\
\qquad otherwise: 
       return \bl{$(r_{1s} + r_{2s}, f_{alt}(f_{1s}, f_{2s}))$}\\
\end{tabular}
\end{center}

\small
\begin{center}
\begin{tabular}{l@{\hspace{1mm}}l}
\bl{$f_{alt}(f_1, f_2) \dn$}\\
\qquad \bl{$\lambda v.\,$} case \bl{$v = Left(v')$}: 
      & return \bl{$Left(f_1(v'))$}\\
\qquad \phantom{$\lambda v.\,$} case \bl{$v = Right(v')$}: 
      & return \bl{$Right(f_2(v'))$}\\      
\end{tabular}
\end{center}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}[c]

{\footnotesize\lstinputlisting[language=Scala,numbers=none,
xleftmargin=-5mm] {../progs/app05.scala}}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Rectification}

\begin{center}
\begin{tabular}{@{\hspace{-3mm}}l}
\bl{$simp(r)$}:\ldots\\
\quad case \bl{$r = r_1 \cdot r_2$}\\
\qquad let \bl{$(r_{1s}, f_{1s}) = simp(r_1)$}\\
\qquad \phantom{let} \bl{$(r_{2s}, f_{2s}) = simp(r_2)$}\smallskip\\
\qquad case \bl{$r_{1s} = \ZERO$}: 
       return \bl{$(\ZERO, f_{error})$}\\
\qquad case \bl{$r_{2s} = \ZERO$}: 
       return \bl{$(\ZERO, f_{error})$}\\
\qquad case \bl{$r_{1s} = \ONE$}: 
return \bl{$(r_{2s}, \lambda v. \,Seq(f_{1s}(Empty), f_{2s}(v)))$}\\
\qquad case \bl{$r_{2s} = \ONE$}: 
return \bl{$(r_{1s}, \lambda v. \,Seq(f_{1s}(v), f_{2s}(Empty)))$}\\
\qquad otherwise: 
       return \bl{$(r_{1s} \cdot r_{2s}, f_{seq}(f_{1s}, f_{2s}))$}\\
\end{tabular}
\end{center}

\small
\begin{center}
\begin{tabular}{l@{\hspace{1mm}}l}
\bl{$f_{seq}(f_1, f_2) \dn$}\\
\qquad \bl{$\lambda v.\,$ case $v = Seq(v_1, v_2)$}: 
      & return \bl{$Seq(f_1(v_1), f_2(v_2))$}\\
\end{tabular}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\begin{frame}[c]

{\footnotesize\lstinputlisting[language=Scala,numbers=none,
xleftmargin=-5mm] {../progs/app06.scala}}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Rectification Example}

\begin{center}
\bl{$\only<1>{(b \cdot c)}%
     \only<2-3>{(\underline{b \cdot c})}%
     \only<1-3>{+}% 
     \only<1>{(\ZERO + \ONE)}%
     \only<2-3>{(\underline{\ZERO + \ONE})}$}%
\only<4->{%
\bl{$\underline{(b \cdot c) + (\ZERO + \ONE)}$}%
}
$\mapsto$
\bl{$(b \cdot c) + \ONE$}
\end{center}\bigskip

\onslide<3->{%
\begin{center}
\begin{tabular}{lcl}
\bl{$f_{s1}$} & \bl{$=$} & \bl{$\lambda v.v$}\\
\bl{$f_{s2}$} & \bl{$=$} & \bl{$\lambda v. \textit{Right}(v)$}
\end{tabular}
\end{center}}

\only<4>{%
\begin{center}
\begin{tabular}{@{}l@{\hspace{1mm}}l@{}}
\bl{$f_{alt}(f_{s1}, f_{s2}) \dn$}\\
\quad \bl{$\lambda v.\,$} 
        case \bl{$v = Left(v')$}: 
      & return \bl{$Left(f_{s1}(v'))$}\\
\quad \phantom{$\lambda v.\,$} 
        case \bl{$v = Right(v')$}: 
      & return \bl{$Right(f_{s2}(v'))$}\\ 
\end{tabular}
\end{center}}%
\only<5->{%
\begin{center}
\begin{tabular}{@{}l@{\hspace{1mm}}l@{}}
\only<5->{\phantom{\bl{$f_{alt}(f_{s1}, f_{s2}) \dn$}}}\\
\quad \bl{$\lambda v.\,$} 
        case \bl{$v = Left(v')$}: 
      & return \bl{$Left(v')$}\\
\quad \phantom{$\lambda v.\,$} 
        case \bl{$v = Right(v')$}: 
      & return \bl{$Right(Right(v'))$}\\ 
\end{tabular}
\end{center}}%

\only<6->{%
\begin{center}
\begin{tabular}{@{}l@{\hspace{4mm}}l@{}}
\bl{$\textit{mkeps}$} simplified case: &
\bl{$\textit{Right}(\textit{Empty})$}\\
rectified case: &
\bl{$\textit{Right}(\textit{Right}(\textit{Empty}))$}
\end{tabular}
\end{center}}%

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Lexing with Simplification}

\begin{center}
\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
  \bl{$lex\,r\,[]$} & \bl{$\dn$}  & \bl{if $nullable(r)$ then $mkeps(r)$ else $error$}\\
  \bl{$lex\,r\,c::s$} & \bl{$\dn$}  & \bl{let $(r', frect) = simp(der(c, r))$}\smallskip\\
                      & & \bl{$inj\,r\,c\,(frect(lex(r', s)))$}\\
\end{tabular}
\end{center}\bigskip

\begin{center}\small
\begin{tikzpicture}[node distance=1.1cm,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{center}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Environments}

Obtaining the ``recorded'' parts of a value: 

\begin{center}
\begin{tabular}{lcl}
  \bl{$env(Empty)$}     & \bl{$\dn$} & \bl{$[]$}\\
  \bl{$env(Char(c))$}   & \bl{$\dn$} & \bl{$[]$}\\
  \bl{$env(Left(v))$}   & \bl{$\dn$} & \bl{$env(v)$}\\
  \bl{$env(Right(v))$}  & \bl{$\dn$} & \bl{$env(v)$}\\
  \bl{$env(Seq(v_1,v_2))$}& \bl{$\dn$} & \bl{$env(v_1) \,@\, env(v_2)$}\\
  \bl{$env(Stars\,[v_1,\ldots ,v_n])$} & \bl{$\dn$} & 
     \bl{$env(v_1) \,@\ldots @\, env(v_n)$}\\
  \bl{$env(Rec(x:v))$} & \bl{$\dn$} & \bl{$(x:|v|) :: env(v)$}\\
\end{tabular}
\end{center}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{While Tokens}

\begin{center}
\begin{tabular}{@{}r@{\hspace{2mm}}c@{\hspace{2mm}}l@{}}
\pcode{WHILE\_REGS} & $\dn$ & \raisebox{-1mm}{\large(}\pcode{("k" : KEYWORD)} +\\ 
                  &       & \phantom{(}\pcode{("i" : ID)} +\\ 
                  &       & \phantom{(}\pcode{("o" : OP)} + \\
                  &       & \phantom{(}\pcode{("n" : NUM)} + \\
                  &       & \phantom{(}\pcode{("s" : SEMI)} +\\ 
                  &       & \phantom{(}\pcode{("p" : (LPAREN + RPAREN))} +\\ 
                  &       & \phantom{(}\pcode{("b" : (BEGIN + END))} +\\ 
                  &       & \phantom{(}\pcode{("w" : WHITESPACE)}\raisebox{-1mm}{\large)$^*$}
\end{tabular}
\end{center}

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



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]

\consolas
\begin{center}
\code{"if true then then 42 else +"}
\end{center}

\only<1>{
\small\begin{tabular}{l}
KEYWORD(if),\\ 
WHITESPACE,\\ 
IDENT(true),\\ 
WHITESPACE,\\ 
KEYWORD(then),\\ 
WHITESPACE,\\ 
KEYWORD(then),\\ 
WHITESPACE,\\ 
NUM(42),\\ 
WHITESPACE,\\ 
KEYWORD(else),\\ 
WHITESPACE,\\ 
OP(+)
\end{tabular}}

\only<2>{
\small\begin{tabular}{l}
KEYWORD(if),\\ 
IDENT(true),\\ 
KEYWORD(then),\\ 
KEYWORD(then),\\ 
NUM(42),\\ 
KEYWORD(else),\\ 
OP(+)
\end{tabular}}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Lexer: Two Rules}

\begin{itemize}
\item Longest match rule (``maximal munch rule''): The 
longest initial substring matched by any regular expression is taken
as next token.\bigskip

\item Rule priority:
For a particular longest initial substring, the first regular
expression that can match determines the token.

\end{itemize}

%\url{http://www.technologyreview.com/tr10/?year=2011}
  
%finite deterministic automata/ nondeterministic automaton

%\item problem with infix operations, for example i-12


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



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
\bl{$zeroable(\ZERO)$}      & \bl{$\dn$} & \bl{$true$}\\
\bl{$zeroable(\ONE)$}         & \bl{$\dn$} & \bl{$\textit{false}$}\\
\bl{$zeroable(c)$}                & \bl{$\dn$} & \bl{$\textit{false}$}\\
\bl{$zeroable(r_1 + r_2)$}        & \bl{$\dn$} & \bl{$zeroable(r_1) \wedge zeroable(r_2)$}\\ 
\bl{$zeroable(r_1 \cdot r_2)$}    & \bl{$\dn$} & \bl{$zeroable(r_1) \vee zeroable(r_2)$}\\
\bl{$zeroable(r^*)$}              & \bl{$\dn$} & \bl{$\textit{false}$}\\
\end{tabular}
\end{center}

\begin{center}
\bl{$zeroable(r)$} if and only if \bl{$L(r) = \{\}$}
\end{center}


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


\end{document}

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