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

\begin{document}

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\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}

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\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},

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    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},

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\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, Strings}

\begin{itemize}

\item \alert{\bf Strings} are lists of characters, for example

\begin{center}

\bl{$[]$},\;\bl{$abc$}  \hspace{2cm}(Pattern match: \bl{$c\!::\!s$})

\end{center}\bigskip

\item A \alert{\bf language} is a set of strings, for example\medskip

\begin{center}

\bl{$\{[], hello, \textit{foobar}, a, abc\}$}

\end{center}\bigskip

\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}

%\item The \alert{\bf meaning} of a regular expression is a set of 

%  strings, or language.

\end{itemize}  

\end{frame}

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\begin{frame}[t]

\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}

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}

\end{frame}

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\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{$\bigcup_{n \ge 0} L(r)^n$}\\

  \end{tabular}\bigskip

\only<2->{  

\hspace{5mm}\textcolor{blue}{$L(r)^0 \;\dn\; \{[]\}$}\\

\textcolor{blue}{$L(r)^{n+1} \;\dn\; L(r) \,@\, L(r)^n$}}  

\end{textblock}

\only<1->{

\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}

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\begin{frame}[t]

\frametitle{\begin{tabular}{c}Concrete Equivalences\end{tabular}}

\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}

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\begin{frame}[t]

\frametitle{\begin{tabular}{c}Corner Cases\end{tabular}}

\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}

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\begin{frame}[t]

\frametitle{\begin{tabular}{c}Simplification Rules\end{tabular}}

\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}

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\newcommand{\YES}{\textcolor{gray}{yes}}

\newcommand{\NO}{\textcolor{gray}{no}}

\newcommand{\FORALLR}{\textcolor{gray}{$\forall$ r.}}

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\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}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}

\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}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Evil Regular Expressions\end{tabular}}

\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}}

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\mode<presentation>{

\begin{frame}[t]

\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}

\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{$f\!\/alse$}\\

\bl{$nullable(\epsilon)$}           & \bl{$\dn$} &  \bl{$true$}\\

\bl{$nullable (c)$}                    & \bl{$\dn$} &  \bl{$f\!alse$}\\

\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{$true$} \\

\end{tabular}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}

\large

If \bl{$r$} matches the string \bl{$c\!::\!s$}, what is a regular 

expression that matches \bl{$s$}?\bigskip\bigskip\bigskip\bigskip

\small

\bl{$der\,c\,r$} gives the answer, Brzozowski 1964

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}The Derivative of a Rexp (2)\end{tabular}}

\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{$(der\,c\,r) \cdot (r^*)$} &\medskip\\\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}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Examples\end{tabular}}

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}}

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\mode<presentation>{

\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\\

        & $\Rightarrow true \,\text{or}\, \textit{false}$\\        

\end{tabular}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}

\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}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}A Problem\end{tabular}}

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}