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

\hfuzz=220pt 

\lstset{language=Scala,
        style=mystyle,
        numbersep=0pt,
        numbers=none,
        xleftmargin=0mm}

\pgfplotsset{compat=1.11}

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

% beamer stuff 
\renewcommand{\slidecaption}{AFL 02, King's College London}


\begin{document}

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

  \normalsize
  \begin{center}
  \begin{tabular}{ll}
  Email:  & christian.urban at kcl.ac.uk\\
  Office: & S1.27 (1st floor Strand Building)\\
  Slides: & KEATS 
  \end{tabular}
  \end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\begin{tabular}{@ {}c@ {}}
    An Efficient Regular\\[-1mm] 
    Expression Matcher\end{tabular}}

\footnotesize
\begin{center}
\begin{tabular}{@{}cc@{}}
\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=4.5cm,
    height=4.5cm, 
    legend entries={Python,Ruby},  
    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};  
\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}
\begin{axis}[
    xlabel={\pcode{a}s},
    ylabel={time in secs},
    enlargelimits=false,
    xtick={0,3000,...,12000},
    xmax=12000,
    ymax=35,
    ytick={0,5,...,30},
    scaled ticks=false,
    axis lines=left,
    width=5.5cm,
    height=4.5cm
]
\addplot[green,mark=square*,mark options={fill=white}] table {re2b.data};
\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
\end{axis}
\end{tikzpicture}
\end{tabular}
\end{center}


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

\begin{itemize}

\item A \alert{\bf language} is a set of strings, for example\medskip
\begin{center}
\bl{$\{[], hello, \textit{foobar}, a, abc\}$}
\end{center}\bigskip

\item \alert{\bf Concatenation} of strings and languages

\begin{center}
\begin{tabular}{rcl}
\bl{$\textit{foo}\;@\;bar$} & \bl{$=$} & \bl{$\textit{foobar}$}\medskip\\
\bl{$A\;@\;B$} & \bl{$\dn$} & \bl{$\{ s_1\,@\,s_2 \;\mid\; s_1 \in A \wedge s_2 \in B\}$}
\end{tabular}
\end{center}
\bigskip

\small
\item [] For example \bl{$A = \{foo, bar\}$}, \bl{$B = \{a, b\}$}

\[
\bl{A \,@\, B = \{fooa, foob, bara, barb\}}
\]

\end{itemize}  

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Power Operation}

\begin{itemize}
\item The \alert{\bf Power} of a language:

\begin{center}
\begin{tabular}{lcl}
\bl{$A^0$}     & \bl{$\dn$} & \bl{$\{[]\}$}\\
\bl{$A^{n+1}$} & \bl{$\dn$} & \bl{$A \,@\, A^n$}
\end{tabular}
\end{center}\bigskip

\item[] For example

\begin{center}
\begin{tabular}{l}
\bl{$A^4 = A \,@\, A \,@\, A \,@\, A$}\\
\bl{$A^0 \dn \{[]\}$}\\
\end{tabular}
\end{center}

\end{itemize}  

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Homework Question}

\begin{itemize}
\item Say \bl{$A = \{[a],[b],[c],[d]\}$}.\bigskip

\begin{center}
How many strings are in \bl{$A^4$}?
\end{center}\bigskip\medskip\pause


\begin{center}
\begin{tabular}{l}
What if \bl{$A = \{[a],[b],[c],[]\}$};\\ 
how many strings are then in \bl{$A^4$}?
\end{tabular}
\end{center}
\end{itemize}  

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Star Operation}

\begin{itemize}
\item The \alert{\bf Star} of a \underline{language}:
\bigskip

\begin{center}
\begin{tabular}{c}
\bl{$A^* \dn \bigcup_{0\le n} A^n$}
\end{tabular}
\end{center}\bigskip

\item[] This expands to 

\[
\bl{A^0 \cup A^1 \cup A^2 \cup A^3 \cup A^4 \cup \ldots}
\]

\small
\[
\bl{\{[]\} \;\cup\; A \;\cup\; A\,@\,A \;\cup\; 
  A\,@\,A\,@\,A \;\cup\; A\,@\,A\,@\,A\,@\,A \cup \ldots}
\]

\end{itemize}  

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Semantic Derivative}

\begin{itemize}
\item The \alert{\bf Semantic Derivative} of a 
\underline{language}\\ wrt to a character \bl{$c$}:
\bigskip

\begin{center}
\bl{$Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } 
\end{center}\bigskip\bigskip

For \bl{$A = \{\textit{foo}, \textit{bar}, \textit{frak}\}$} then

\begin{center}
\bl{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}
$Der\,f\,A$ & $=$ & $\{\textit{oo}, \textit{rak}\}$\\
$Der\,b\,A$ & $=$ &  $\{\textit{ar}\}$\\  
$Der\,a\,A$ & $=$ & $\varnothing$\\
\end{tabular}}
\end{center}

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

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

Their inductive definition:


\begin{textblock}{6}(2,7.5)
  \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
  \bl{$r$} & \bl{$::=$}  & \bl{$\varnothing$}  & null\\
         & \bl{$\mid$} & \bl{$\epsilon$}       & empty string / \pcode{""} / $[]$\\
         & \bl{$\mid$} & \bl{$c$}              & character\\
         & \bl{$\mid$} & \bl{$r_1 \cdot r_2$}  & sequence\\
         & \bl{$\mid$} & \bl{$r_1 + r_2$}      & alternative / choice\\
         & \bl{$\mid$} & \bl{$r^*$}            & star (zero or more)\\
  \end{tabular}
\end{textblock}
  
\only<2->{\footnotesize
\begin{textblock}{9}(2,0.5)
\begin{bubble}[9.8cm]
\lstinputlisting{../progs/app01.scala}
\end{bubble}
\end{textblock}}  
  
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\begin{tabular}{c}The Meaning of a\\[-2mm] Regular Expression\end{tabular}}

\begin{textblock}{15}(1,4)
 \begin{tabular}{@ {}rcl}
 \bl{$L(\varnothing)$}     & \bl{$\dn$} & \bl{$\varnothing$}\\
 \bl{$L(\epsilon)$}        & \bl{$\dn$} & \bl{$\{[]\}$}\\
 \bl{$L(c)$}               & \bl{$\dn$} & \bl{$\{[c]\}$}\\
 \bl{$L(r_1 + r_2)$}       & \bl{$\dn$} & \bl{$L(r_1) \cup L(r_2)$}\\
 \bl{$L(r_1 \cdot r_2)$}   & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\
 \bl{$L(r^*)$}             & \bl{$\dn$} & \bl{$(L(r))^*$}\\
 \end{tabular}
\end{textblock}

\begin{textblock}{6}(9,12)\small
\bl{$L$} is a function from regular expressions to 
sets of strings\\
\bl{$L$ : Rexp $\Rightarrow$ Set$[$String$]$}
\end{textblock}

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

\large
\begin{center}
What is \bl{$L(a^*)$}?
\end{center}  
  
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\begin{tabular}{c}
 When Are Two Regular\\[-1mm]
 Expressions Equivalent?\end{tabular}}
\large
\begin{center}
\bl{$r_1 \equiv r_2 \;\;\dn\;\; L(r_1) = L(r_2)$}
\end{center}  
  
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Concrete Equivalences}

\begin{center}
\bl{\begin{tabular}{rcl}
$(a + b)  + c$ & $\equiv$ & $a + (b + c)$\\
$a + a$        & $\equiv$ & $a$\\
$a + b$        & $\equiv$ & $b + a$\\
$(a \cdot b) \cdot c$ & $\equiv$ & $a \cdot (b \cdot c)$\\
$c \cdot (a + b)$ & $\equiv$ & $(c \cdot a) + (c \cdot b)$\bigskip\bigskip\\\pause
$a \cdot a$       & $\not\equiv$ & $a$\\
$a + (b \cdot c)$ & $\not\equiv$ & $(a + b) \cdot (a + c)$\\
\end{tabular}}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Corner Cases}

\begin{center}
\bl{\begin{tabular}{rcl}
$a \cdot \varnothing$ & $\not\equiv$ & $a$\\
$a + \epsilon$ & $\not\equiv$ & $a$\\
$\epsilon$ & $\equiv$ & $\varnothing^*$\\
$\epsilon^*$ & $\equiv$ & $\epsilon$\\
$\varnothing^*$ & $\not\equiv$ & $\varnothing$\\
\end{tabular}}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Simplification Rules}

\begin{center}
\bl{\begin{tabular}{rcl}
$r + \varnothing$  & $\equiv$ & $r$\\
$\varnothing + r$  & $\equiv$ & $r$\\
$r \cdot \epsilon$ & $\equiv$ & $r$\\
$\epsilon \cdot r$     & $\equiv$ & $r$\\
$r \cdot \varnothing$ & $\equiv$ & $\varnothing$\\
$\varnothing \cdot r$ & $\equiv$ & $\varnothing$\\
$r + r$ & $\equiv$ & $r$
\end{tabular}}
\end{center}

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

%\newcommand{\YES}{\textcolor{gray}{yes}}
%\newcommand{\NO}{\textcolor{gray}{no}}
%\newcommand{\FORALLR}{\textcolor{gray}{$\forall$ r.}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\begin{tabular}{c}The Specification\\[-1mm] for Matching\end{tabular}}

\large
A regular expression \bl{$r$} matches a string \bl{$s$}\\ if and only if

\begin{center}
\bl{$s \in L(r)$}\\ 
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\bl{$(a?\{n\}) \cdot a\{n\}$}}

\begin{center}
\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=9cm,
    height=7cm, 
    legend entries={Python,Ruby},  
    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};  
\end{axis}
\end{tikzpicture}
\end{center}

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

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

\begin{itemize}
\item \alert{R}egular \alert{e}xpression \alert{D}enial \alert{o}f \alert{S}ervice (ReDoS)\bigskip
\item Evil regular expressions\medskip
\begin{itemize}
\item \bl{$(a?\{n\}) \cdot a\{n\}$}
\item \bl{$(a^+)^+$}
\item \bl{$([a$\,-\,$z]^+)^*$}
\item \bl{$(a + a \cdot a)^+$}
\item \bl{$(a + a?)^+$}
\end{itemize}
\end{itemize}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{A Matching Algorithm}


\ldots{}whether a regular expression can match the empty string:
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
\bl{$nullable(\varnothing)$}    & \bl{$\dn$} & \bl{\textit{false}}\\
\bl{$nullable(\epsilon)$}       & \bl{$\dn$} & \bl{\textit{true}}\\
\bl{$nullable (c)$}             & \bl{$\dn$} & \bl{\textit{false}}\\
\bl{$nullable (r_1 + r_2)$}     & \bl{$\dn$} & \bl{$nullable(r_1) \vee nullable(r_2)$} \\ 
\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$} \\
\bl{$nullable (r^*)$}           & \bl{$\dn$} & \bl{\textit{true}}\\
\end{tabular}
\end{center}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Derivative of a Rexp}

\large
If \bl{$r$} matches the string \bl{$c\!::\!s$}, what is a regular 
expression that matches just \bl{$s$}?\bigskip\bigskip\bigskip\bigskip

\small
\bl{$der\,c\,r$} gives the answer, Brzozowski 1964
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Derivative of a Rexp}

\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
  \bl{$der\, c\, (\varnothing)$}      & \bl{$\dn$} & \bl{$\varnothing$} & \\
  \bl{$der\, c\, (\epsilon)$}           & \bl{$\dn$} & \bl{$\varnothing$} & \\
  \bl{$der\, c\, (d)$}                     & \bl{$\dn$} & \bl{if $c = d$ then $\epsilon$ else $\varnothing$} & \\
  \bl{$der\, c\, (r_1 + r_2)$}        & \bl{$\dn$} & \bl{$der\, c\, r_1 + der\, c\, r_2$} & \\
  \bl{$der\, c\, (r_1 \cdot r_2)$}  & \bl{$\dn$}  & \bl{if $nullable (r_1)$}\\
  & & \bl{then $(der\,c\,r_1) \cdot r_2 + der\, c\, r_2$}\\ 
  & & \bl{else $(der\, c\, r_1) \cdot r_2$}\\
  \bl{$der\, c\, (r^*)$}          & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\bigskip\\\pause

  \bl{$\textit{ders}\, []\, r$}     & \bl{$\dn$} & \bl{$r$} & \\
  \bl{$\textit{ders}\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\textit{ders}\,s\,(der\,c\,r)$} & \\
  \end{tabular}
\end{center}

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

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

Given \bl{$r \dn ((a \cdot b) + b)^*$} what is

\begin{center}
\begin{tabular}{l}
\bl{$der\,a\,r =\;?$}\\
\bl{$der\,b\,r =\;?$}\\
\bl{$der\,c\,r =\;?$}
\end{tabular}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{The Algorithm}

\begin{center}
\begin{tabular}{@{}rll@{}}
Input: & \bl{$r_1$}, \bl{$abc$}\medskip\\
Step 1: & build derivative of \bl{$a$} and \bl{$r_1$} & \bl{$(r_2 = der\,a\,r_1)$}\smallskip\\
Step 2: & build derivative of \bl{$b$} and \bl{$r_2$} & \bl{$(r_3 = der\,b\,r_2)$}\smallskip\\
Step 3: & build derivative of \bl{$c$} and \bl{$r_3$} & \bl{$(r_4 = der\,b\,r_3)$}\smallskip\\
Step 4: & the string is exhausted; test & (\bl{$nullable(r_4)$})\\
        & whether \bl{$r_4$} can recognise\\
        & the empty string\smallskip\\
Output: & result of the test\\
        & $\Rightarrow \bl{\textit{true}} \,\text{or}\, 
                       \bl{\textit{false}}$\\        
\end{tabular}
\end{center}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Idea of the Algorithm}

If we want to recognise the string \bl{$abc$} with regular 
expression \bl{$r_1$} then\medskip

\begin{enumerate}
\item \bl{$Der\,a\,(L(r_1))$}\pause
\item \bl{$Der\,b\,(Der\,a\,(L(r_1)))$}\pause
\item \bl{$Der\,c\,(Der\,b\,(Der\,a\,(L(r_1))))$}\bigskip
\item finally we test whether the empty string is in this set\medskip
\end{enumerate}

The matching algorithm works similarly, just over regular expressions instead of sets.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\bl{$(a?\{n\}) \cdot a\{n\}$}}

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    xlabel={\pcode{a}s},
    ylabel={time in secs},
    enlargelimits=false,
    xtick={0,5,...,30},
    xmax=30,
    ytick={0,5,...,30},
    scaled ticks=false,
    axis lines=left,
    width=7cm,
    height=7cm, 
    legend entries={Python,Ruby,Scala V1},  
    legend pos=outer north east,
    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 {re1.data};  
\end{axis}
\end{tikzpicture}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{A Problem}

We represented the ``n-times'' \bl{$a\{n\}$} as a 
sequence regular expression:

\begin{center}
\begin{tabular}{rl}
1: & \bl{$a$}\\
2: & \bl{$a\cdot a$}\\
3: & \bl{$a\cdot a\cdot a$}\\
& \ldots\\
13: & \bl{$a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a$}\\
& \ldots\\
20:
\end{tabular}
\end{center}

This problem is aggravated with \bl{$a?$} being represented 
as \bl{$\epsilon + a$}.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Solving the Problem}

What happens if we extend our regular expressions

\begin{center}
\begin{tabular}{rcl}
\bl{$r$} & \bl{$::=$} & \bl{\ldots}\\
             & \bl{$\mid$} & \bl{$r\{n\}$}\\
             & \bl{$\mid$} & \bl{$r?$} 
\end{tabular}
\end{center}

What is their meaning? What are the cases for \bl{$nullable$} and \bl{$der$}?
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


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

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    xlabel={\pcode{a}s},
    ylabel={time in secs},
    enlargelimits=false,
    xtick={0,200,...,1000},
    xmax=1000,
    ytick={0,5,...,30},
    scaled ticks=false,
    axis lines=left,
    width=9.5cm,
    height=7cm, 
    legend entries={Python,Ruby,Scala V1,Scala V2},  
    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 {re1.data};  
\addplot[green,mark=square*,mark options={fill=white}] 
  table {re2b.data};
\end{axis}
\end{tikzpicture}
\end{center}

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


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

Recall the example of \bl{$r \dn ((a \cdot b) + b)^*$} with

\begin{center}
\begin{tabular}{l}
\bl{$der\,a\,r = ((\epsilon \cdot b) + \varnothing) \cdot r$}\\
\bl{$der\,b\,r = ((\varnothing \cdot b) + \epsilon)\cdot r$}\\
\bl{$der\,c\,r = ((\varnothing \cdot b) + \varnothing)\cdot r$}
\end{tabular}
\end{center}

What are these regular expressions equivalent to?

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



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

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    xlabel={\pcode{a}s},
    ylabel={time in secs},
    enlargelimits=false,
    xtick={0,3000,...,12000},
    xmax=12000,
    ymax=35,
    ytick={0,5,...,30},
    scaled ticks=false,
    axis lines=left,
    width=9cm,
    height=7cm
]
\addplot[green,mark=square*,mark options={fill=white}] table {re2b.data};
\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
\end{axis}
\end{tikzpicture}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Proofs about Rexps}

Remember their inductive definition:\\[5cm]

\begin{textblock}{6}(5,5)
  \begin{tabular}{@ {}rrl}
  \bl{$r$} & \bl{$::=$}  & \bl{$\varnothing$}\\
         & \bl{$\mid$} & \bl{$\epsilon$}     \\
         & \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{textblock}

If we want to prove something, say a property \bl{$P(r)$}, for all regular expressions \bl{$r$} then \ldots

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Proofs about Rexp (2)}

\begin{itemize}
\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
\item \bl{$P$} holds for \bl{$r_1 + r_2$} under the assumption that \bl{$P$} already
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
\item \bl{$P$} holds for \bl{$r_1 \cdot r_2$} under the assumption that \bl{$P$} already
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
\item \bl{$P$} holds for \bl{$r^*$} under the assumption that \bl{$P$} already
holds for \bl{$r$}.
\end{itemize}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Proofs about Rexp (3)}

Assume \bl{$P(r)$} is the property:

\begin{center}
\bl{$nullable(r)$} if and only if \bl{$[] \in L(r)$}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Proofs about Rexp (4)}

\begin{center}
\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
$rev(\varnothing)$   & $\dn$ & $\varnothing$\\
$rev(\epsilon)$         & $\dn$ & $\epsilon$\\
$rev(c)$                      & $\dn$ & $c$\\
$rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\
$rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
$rev(r^*)$                   & $\dn$ & $rev(r)^*$\\
\end{tabular}}
\end{center}


We can prove

\begin{center}
\bl{$L(rev(r)) = \{s^{-1} \;|\; s \in L(r)\}$}
\end{center}

by induction on \bl{$r$}.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Proofs about Rexp (5)}

Let \bl{$Der\,c\,A$} be the set defined as

\begin{center}
\bl{$Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } 
\end{center}

We can prove

\begin{center}
\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
\end{center}

by induction on \bl{$r$}.

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Proofs about Strings\end{tabular}}

If we want to prove something, say a property \bl{$P(s)$}, for all strings \bl{$s$} then \ldots\bigskip

\begin{itemize}
\item \bl{$P$} holds for the empty string, and\medskip
\item \bl{$P$} holds for the string \bl{$c\!::\!s$} under the assumption that \bl{$P$}
already holds for \bl{$s$}
\end{itemize}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Proofs about Strings (2)\end{tabular}}

We can finally prove

\begin{center}
\bl{$matches(r, s)$}  if and only if  \bl{$s \in L(r)$} 
\end{center}


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

\end{document}

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