% !TEX program = xelatex
\documentclass[dvipsnames,14pt,t,xelatex,aspectratio=169,xcolor={table}]{beamer}
\usepackage{../slides}
\usepackage{../graphics}
\usepackage{../langs}
\usepackage{../data}

\hfuzz=220pt 

\pgfplotsset{compat=1.11}


\usepackage{tcolorbox}
\newtcolorbox{mybox}{colback=red!5!white,colframe=red!75!black}
\newtcolorbox{mybox2}[1]{colback=red!5!white,colframe=red!75!black,fonttitle=\bfseries,title=#1}
\newtcolorbox{mybox3}[1]{colback=Cyan!5!white,colframe=Cyan!75!black,fonttitle=\bfseries,title=#1}

% a hand written lexer for SML
% https://ponyo.org/
% https://github.com/eatonphil/ponyo/blob/master/src/Sml/Lexer.sml
\beamertemplateballitem

\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\\[3mm] 
  \end{tabular}}

  \normalsize
  \begin{center}
  \begin{tabular}{ll}
    Email:  & christian.urban at kcl.ac.uk\\
    %Office Hours: & Thursdays 12 -- 14\\
    %Location: & N7.07 (North Wing, Bush House)\\
    Slides \& Progs: & KEATS (also homework is there)\\  
  \end{tabular}
  \end{center}


  \begin{center}
    \begin{tikzpicture}
      \node[drop shadow,fill=white,inner sep=0pt] 
      {\footnotesize\rowcolors{1}{capri!10}{white}
        \begin{tabular}{|p{4.8cm}|p{4.8cm}|}\hline
          1 Introduction, Languages          & 6 While-Language \\
          2 Regular Expressions, Derivatives & 7 Compilation, JVM \\
          3 Automata, Regular Languages      & 8 Compiling Functional Languages \\
          \cellcolor{blue!50}
          4 Lexing, Tokenising               & 9 Optimisations \\
          5 Grammars, Parsing                & 10 LLVM \\ \hline
        \end{tabular}%
      };
    \end{tikzpicture}
  \end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[c]
%\frametitle{Coursework}
%
%\begin{itemize}
%\item \bl{$\der\,c\,(r^+) \dn \der\,c (r\cdot r^*)$}\quad given
%that \bl{$r^+ \dn r\cdot r^*$}
%\end{itemize}\bigskip\pause

%\begin{center}
%\begin{tabular}{lcl}
%\bl{$\der\,c\,(r\cdot r^*)$} & \bl{$\dn$} & 
%\only<2-4>{if \bl{$nullable\,r$}}%
%\only<5>{\bl{$(\der\,c\,r)\cdot r^*$}}\\
% & & 
% \only<2>{then \bl{$(\der\,c\,r)\cdot r^* \,+\, \der\,c\,(r^*)$}}%
% \only<3>{then \bl{$(\der\,c\,r)\cdot r^* \,+\, (\der\,c\,r)\cdot r^*$}}%
% \only<4>{then \bl{$(\der\,c\,r)\cdot r^*$}}\\
% & & \only<2-4>{else \bl{$(\der\,c\,r)\cdot r^*$}}
%\end{tabular}  
%\end{center}  
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[c]
%  \frametitle{Coursework (2)}
%  
%  \begin{itemize}
%  \item \bl{\texttt{CFUN(f: Char => Boolean)}}
%  \end{itemize}\medskip
%
%  \begin{center}
%  \begin{tabular}{l}
%  \bl{\texttt{CHAR(c: Char)}} \bl{$\dn$}\\
%     \quad\bl{\texttt{CFUN(\_ == c)}}\medskip\\
%  \bl{\texttt{RANGE(cs: Set[Char])}} \bl{$\dn$}\\
%     \quad\bl{\texttt{CFUN(cs.contains(\_))}}\medskip\\
%  \bl{\texttt{ALL}} \bl{$\dn$}\\
%     \quad\bl{\texttt{CFUN((c: Char) => true)}}\\
%  \end{tabular}  
%  \end{center}  
%\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,drop shadow}]

  \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 operands, 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/while-tests/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\bigskip

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

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

\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]

\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<1->{
\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{$|Stars\,[v_1,\ldots ,v_n]|$} & \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{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}
\large

  \begin{center}
\begin{tikzpicture}[scale=3,node distance=1.2cm,
                    every node/.style={minimum size=7mm}]
\node (r)  {$r$};
\node (rd) [right=of r]{$r_{der}$};
\draw[->,line width=1mm](r)--(rd) node[above,midway] {$\der\,c$};
\node (vd) [below=of r2]{$v_{der}$};
\draw[->,line width=1mm](rd) -- (vd);
\node (v) [left=of vd] {$v$};
\draw[->,line width=1mm](vd)--(v) node[below,midway] {$inj\,c$};
\draw[->,line width=0.5mm,dotted](r) -- (v) node[left,midway,red] {\bf ?};
\end{tikzpicture}
\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
  \begin{tabular}{l@{\hspace{2mm}}l}
  \bl{$inj$}: & 1st arg $\mapsto$ a rexp; 2nd arg $\mapsto$ a character; 
  3rd arg $\mapsto$ a value\\
              & result $\mapsto$ a value 
  \end{tabular}
  \end{frame}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{frame}[t]
  
  \begin{center}
  \begin{tabular}{@{\hspace{-3mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
    \bl{$inj\,(c)\,c\,(Empty)$} & \bl{$\dn$}  & \bl{$Char\,c$}\\
  \end{tabular}
  \end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{frame}[t]
  
  \begin{center}
    \begin{tabular}{@{\hspace{-3mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
    \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{$Right(inj\,r_2\,c\,v)$}\\

  \end{tabular}
  \end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{frame}[t]
  
  \begin{center}
    \begin{tabular}{@{\hspace{-8mm}}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}}
    \\[-15mm]  
    \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{$Seq(mkeps(r_1),inj\,r_2\,c\,v)$}\\  
  \end{tabular}
\end{center}


\begin{textblock}{13}(1,13.4)
\begin{bubble}[13cm]
\small
\bl{$\der\, c\, (r_1 \cdot r_2)$} \bl{$\dn$} \bl{\textbf{if} $nullable (r_1)$}
 \bl{\textbf{then} $(\der\,c\,r_1) \cdot r_2 + \der\, c\, r_2$}
 \bl{\textbf{else} $(\der\, c\, r_1) \cdot r_2$}
\end{bubble}
\end{textblock}

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

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

\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{$lex\,r\,a::s$} & \bl{$\dn$}  & \bl{$inj\,r\,a\,lex(der(a,r), s)$}\\
\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{$der\,c\,(x:r) \dn der\,c\,r$}
\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{$(z: ((x:ab) + (y:ba))$}

\begin{textblock}{6}(10,10)
\begin{bubble}[2cm]
\small
\begin{tabular}{l}
  \bl{$(id : r_{id})$}\\
  \bl{$(key : r_{key})$}\\
\end{tabular}
\end{bubble}
\end{textblock}


\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\pause
\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 $\_ \cdot \_$}

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


  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[c]
%\frametitle{\begin{tabular}{c}Last Week\\[-2mm] 
%            Regexes and Values\end{tabular}}
%
%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{$Stars [v_1,\ldots\,v_n]$} \\
%  \end{tabular}
%\end{column}
%\end{columns}
%\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{Simplification}
%
%\begin{itemize}
%\item If we simplify after the derivative, then we are builing the
%value for the simplified regular expression, but \emph{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{$(b \cdot c) + (\ZERO + \ONE)$}
%$\mapsto$
%\bl{$(b \cdot c) + \ONE$}
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[t]

% \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{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{$der\,c\,(x:r) \dn der\,c\,r$}
%\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]
\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{Coursework: PLs (16)}
%
%\begin{itemize}
%\item Java (16)
%\item C++, C, C\# (14)
%\item JavaScript (10)
%\item Scala (9)
%\item Python (9)  
%\item PHP (6)
%\item Haskell (3)
%\item Ruby (4)
%\item Bash, Perl, Powershell (2)
%\item TypeScript (1)
%\item R (1)
%\item Coconut (1)  
%\item Pascal (1)
%\end{itemize}  
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[c]
%\frametitle{Coursework: Nullable}
%
%\begin{center}
%\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
%  \bl{$nullable([c_1 c_2 \ldots c_n])$}  & \bl{$\dn$} & $?$\\
%  \bl{$nullable(r^+)$}                   & \bl{$\dn$} & $?$\\
%  \bl{$nullable(r^?)$}                   & \bl{$\dn$} & $?$\\
%  \bl{$nullable(r^{\{n\}})$}              & \bl{$\dn$} & $?$\\
%  \bl{$nullable(r^{\{n..\}})$}            & \bl{$\dn$} & $?$\\
%  \bl{$nullable(r^{\{..n\}})$}            & \bl{$\dn$} & $?$\\
%  \bl{$nullable(r^{\{n..m\}})$}           & \bl{$\dn$} & $?$\\
%  \bl{$nullable(\sim{}r)$}               & \bl{$\dn$} & $?$\\
%                                                        
%\end{tabular}
%\end{center}
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[c]
%%%\frametitle{Coursework: der}
%
%\begin{center}
%\begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
%  \bl{$der\, c\, ([c_1 c_2 \ldots c_n])$}  & \bl{$\dn$} & $?$\\
%  \bl{$der\, c\, (r^+)$}                   & \bl{$\dn$} & $?$\\
%  \bl{$der\, c\, (r^?)$}                   & \bl{$\dn$} & $?$\\
%  \bl{$der\, c\, (r^{\{n\}})$}              & \bl{$\dn$} &
%     \bl{$if\;n=0\;then\;\ZERO\;else\;(der\,c\,r)\cdot r^{\{n-\liningnums{1}\}}$}\\
%  \bl{$der\, c\, (r^{\{n..\}})$}            & \bl{$\dn$} &
%     \bl{$if\;n=0\;then (der\,c\,r)\cdot r^*$}\\
%  & & \bl{$\phantom{if\;n=0\;}else \;(der\,c\,r)\cdot r^{\{n-\liningnums{1}..\}}$}\\
%  \bl{$der\, c\, (r^{\{..n\}})$}            & \bl{$\dn$} &
%     \bl{$if\;n=0\;then\;\ZERO\;else\;(der\,c\,r)\cdot r^{\{..n-\liningnums{1}\}}$}\\
%  
%  \bl{$der\, c\, (r^{\{n..m\}})$}          & \bl{$\dn$} &
%     \bl{$if\;n = 0 \wedge m = 0\;then\;\ZERO\; else$}\\
%  \multicolumn{3}{l}{\bl{$if\;n = 0 \wedge m > 0\;then\;(der\,c\,r)\cdot r^{\{..m-\liningnums{1}\}}$}}\\
%  \multicolumn{3}{l}{\bl{$\phantom{if\;n = 0 \wedge m > 0\;}else
%          \;(der\,c\,r)\cdot r^{\{n-\liningnums{1}..m-\liningnums{1}\}}$}}\\
%  \bl{$der\, c\, (\sim{}r)$}              & \bl{$\dn$} & $?$\\
%\end{tabular}
%\end{center}
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[c]
%\frametitle{Coursework: CFUN}
%
%\begin{center}
%\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
%  \bl{$nullable(CFUN(\_))$}  & \bl{$\dn$} & \bl{$false$}\\
%  \bl{$der\,c\,(CFUN(f))$}   & \bl{$\dn$} &
%     \bl{$if\;f(c)\;then\;\ONE\;else\;\ZERO$}\bigskip\\
%  \bl{$CHAR(c)$}                   & \bl{$\dn$} & \bl{$CFUN(\lambda{}d.\;c=d)$}\\
%  \bl{$CSET([c_1,\ldots,c_n])$} & \bl{$\dn$} & \bl{$CFUN(\lambda{}d.\;d\in [c_1,\ldots,c_n])$}\\
%  \bl{$ALL$}                   & \bl{$\dn$} & \bl{$CFUN(\lambda{}d.\;true)$}\\                                                      
%\end{tabular}
%\end{center}
%
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[t]
\begin{mybox3}{}
  Are dfas completed by definition as in do they have a to have transitions
  for every char at every state?
\end{mybox3}
\end{frame}

\begin{frame}[t]
\begin{mybox3}{}
  How can you tell if a language will be regular or irregular?
\end{mybox3}
\end{frame}

\begin{frame}<1-12>[c]
\end{frame}

\begin{frame}[c]
  \small
  \begin{itemize}
\item[$\bullet$] contentwise probably the most enjoyable module I have had so far at KCL 

\item[$\bullet$] I personally found the coursework sheets a bit unclear. For example I couldnt see what was required from the CFUN section but once explained it actually was very easy and didnt take long to get working

\item[$\bullet$] Please can tutorial sessions be recorded \& linked on Keats.
  
\item[$\bullet$] One of the best taught modules I've had, with a genuinely interested and engaging lecturer. Thanks Dr.~Urban!
\end{itemize}
\end{frame}

\begin{frame}[c]
  \small
  \begin{itemize}
\item[$\bullet$] Dr.~Urban is honestly a great lecturer, he's incredibly helpful and responsive. He also teaches at a very good pace and explains things clearly so students who sometimes struggle with the content like myself can keep up. It's a pleasure to learn from him.
  
\item[$\bullet$] I just wish the Coursework content was explained better, in such a way that allows students to get on with the coursework as soon as possible.
  
\item[$\bullet$] I'm thoroughly enjoying the module so far. I find that the handouts and homework really solidify my knowledge and the feedback is extremely useful. I enjoy doing the homework and then using the feedback alongside the workshop to correct where I have gone wrong.
  
\item[$\bullet$] I enjoy this module and think it is taught well. The homework is very useful.
\end{itemize}
\end{frame}

\begin{frame}[c]
  \begin{center}
  \begin{tabular}{cc}  
  \includegraphics[scale=0.3]{s1.png} &
  \includegraphics[scale=0.3]{s2.png} \\
  \includegraphics[scale=0.3]{s3.png} &
  \includegraphics[scale=0.3]{s4.png} \\
  \end{tabular}                                      
  \end{center}
\end{frame}

\begin{frame}[c]
  \begin{center}
  \begin{tabular}{cc}  
  \includegraphics[scale=0.3]{s5.png} &
  \includegraphics[scale=0.3]{s6.png} \\
  \includegraphics[scale=0.3]{s7.png} &
  \includegraphics[scale=0.3]{s8.png} \\
  \end{tabular}                                      
  \end{center}
\end{frame}

\begin{frame}[c]
  \begin{center}
  \begin{tabular}{cc}  
  \includegraphics[scale=0.3]{s9.png} 
  \end{tabular}                                      
  \end{center}
\end{frame}

\end{document}

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