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\usepackage{../graphics}

\usepackage{../langs}

\usepackage{../data}

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\pgfplotsset{compat=1.12}

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

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

\begin{document}

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\begin{frame}[t]

\frametitle{%

  \begin{tabular}{@ {}c@ {}}

  \\[-3mm]

  \LARGE Compilers and \\[-1mm] 

  \LARGE Formal Languages (2)\\[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}

\end{frame}

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\begin{frame}[t]

  \frametitle{

    Lets Implement an Efficient\\[-2mm] 

    Regular Expression Matcher}

\footnotesize

\begin{center}

  {\normalsize Graphs: \bl{$a^{?\{n\}} \cdot a^{\{n\}}$} and strings \bl{$\underbrace{\,a\ldots a\,}_{n}$}}\medskip\\

\begin{tabular}{@{}cc@{}}

\begin{tikzpicture}

\begin{axis}[

    xlabel={$n$},

    x label style={at={(1.05,0.0)}},

    ylabel={time in secs},

    enlargelimits=false,

    xtick={0,5,...,30},

    xmax=33,

    ymax=35,

    ytick={0,5,...,30},

    scaled ticks=false,

    axis lines=left,

    width=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={$n$},

    x label style={at={(1.1,0.05)}},

    ylabel={time in secs},

    enlargelimits=false,

    xtick={0,5000,...,10000},

    xmax=11500,

    ymax=35,

    ytick={0,5,...,30},

    scaled ticks=false,

    axis lines=left,

    width=5cm,

    height=4.5cm,

    legend entries={Scala V2,Scala V3},

    legend pos=north east,

    legend cell align=left]

\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};

\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};

\end{axis}

\end{tikzpicture}

\end{tabular}

\end{center}

\small

In the handouts is a similar graph for \bl{$(a^*)^* \cdot b$} and Java 8, 

JavaScript and Python.

\end{frame}

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%\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}

%

%\item sometimes also called \alert{catastrophic backtracking}\bigskip

%\item \small\ldots I hope you have watched the video by the

%  StackExchange engineer  

%\end{itemize}

%

%\end{frame}

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\begin{frame}[c]

\frametitle{(Basic) Regular Expressions}

Their inductive definition:

\begin{center}

  \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}

  \bl{$r$} & \bl{$::=$}  & \bl{$\ZERO$}  & nothing\\

         & \bl{$\mid$} & \bl{$\ONE$}       & empty string / \pcode{""} / $[]$\\

         & \bl{$\mid$} & \bl{$c$}                         & character\\

         & \bl{$\mid$} & \bl{$r_1 + r_2$}  & alternative / choice\\

         & \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\

         & \bl{$\mid$} & \bl{$r^*$}            & star (zero or more)\\

  \end{tabular}

\end{center}

\vspace{\fill}

Q: \;What about \;\bl{$r \cdot 0$} \; ?

\end{frame}

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\begin{frame}[c]

\frametitle{Languages (Sets of Strings)}

\begin{itemize}

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

\begin{center}

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

\end{center}\bigskip

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

\begin{center}

\begin{tabular}{rcl}

\bl{$foo\;@\;bar$} & \bl{$=$} & \bl{$foobar$}\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}

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\begin{frame}[c]

  \frametitle{Two Corner Cases}

  \Large

  \begin{center}

  \bl{$A \,@\, \{[]\} = \;?$}\bigskip\bigskip\pause\\

  \bl{$A \,@\, \{\} = \;?$}

  \end{center}  

  \end{frame}

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\begin{frame}[c]

\frametitle{

    The Meaning of a\\[-2mm] 

    Regular Expression}

 ...all the strings a regular expression can match.   

\begin{center}

 \begin{tabular}{rcl}

 \bl{$L(\ZERO)$}  & \bl{$\dn$} & \bl{$\{\}$}\\

 \bl{$L(\ONE)$}     & \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{$\dn$} & \\

  \end{tabular}

\end{center}

\begin{textblock}{14}(1.5,13.5)\small

\bl{$L$} is a function from regular expressions to 

sets of strings (languages):\smallskip\\

\bl{\quad$L$ : Rexp $\Rightarrow$ Set$[$String$]$}

\end{textblock}

\end{frame}

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\begin{frame}[c]

\frametitle{The Power Operation}

\begin{itemize}

\item The \alert{\textbf{\boldmath$n$th 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}{lcl@{\hspace{10mm}}l}

\bl{$A^4$} & \bl{$=$} & \bl{$A \,@\, A \,@\, A \,@\, A$} & \bl{$(@\,\{[]\})$}\\

\bl{$A^1$} & \bl{$=$} & \bl{$A$} & \bl{$(@\,\{[]\})$}\\

\bl{$A^0$} & \bl{$=$} & \bl{$\{[]\}$}\\

\end{tabular}

\end{center}

\end{itemize}  

\end{frame}

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\begin{frame}[c]

  \frametitle{Homework Question}

  \begin{itemize}

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

  \item[]

  How many strings are in \bl{$A^4$}\,?

  \bigskip\medskip\pause

 \item[]

  What if \bl{$A = \{[a],[b],[c],[]\}$};\\ 

  how many strings are then in \bl{$A^4$}\,?

  \end{itemize}  

\end{frame}

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\begin{frame}[c]

\frametitle{The Star Operation}

\begin{itemize}

\item The \alert{\bf Kleene Star} of a \underline{language}:

\bigskip

\begin{center}

\begin{tabular}{c}

\bl{$A\star \dn \bigcup_{0\le n} A^n$}

\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}

\]

or

\small

\[

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

  A\,@\,A\,@\,A \;\cup\; A\,@\,A\,@\,A\,@\,A \cup \ldots}

\]

\end{itemize}  

\end{frame}

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\begin{frame}[c]

\frametitle{

    The Meaning of a\\[-2mm] 

    Regular Expression}

...all the strings a regular expression can match.   

\begin{center}

 \begin{tabular}{rcl}

 \bl{$L(\ZERO)$}  & \bl{$\dn$} & \bl{$\{\}$}\\

 \bl{$L(\ONE)$}     & \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{$(L(r))\star \quad\dn \bigcup_{0 \le n} L(r)^n$}\\

  \end{tabular}

\end{center}

%\begin{textblock}{9}(6,12)\small

%\bl{$L$} is a function from regular expressions to 

%sets of strings (languages):\smallskip\\

%\bl{$L$ : Rexp $\Rightarrow$ Set$[$String$]$}

%\end{textblock}

\end{frame}

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\begin{frame}[c]

  \frametitle{

   When Are Two Regular\\[-1mm]

   Expressions Equivalent?}

   \begin{bubble}[10cm]

    \large

    Two regular expressions \bl{$r_1$} and \bl{$r_2$} are equivalent 

    provided:\medskip 

    \begin{center}

      \bl{$r_1 \equiv r_2 \;\;\dn\;\; L(r_1) = L(r_2)$}  

    \end{center}\medskip

    \end{bubble}

\end{frame}

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\begin{frame}[c]

\frametitle{Some 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}

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\begin{frame}[c]

\frametitle{Some Corner Cases}

\begin{center}

\bl{\begin{tabular}{rcl}

$a \cdot \ZERO$ & $\not\equiv$ & $a$\\

$a + \ONE$ & $\not\equiv$ & $a$\\

$\ONE$ & $\equiv$ & $\ZERO^*$\\

$\ONE^*$ & $\equiv$ & $\ONE$\\

$\ZERO^*$ & $\not\equiv$ & $\ZERO$\\

\end{tabular}}

\end{center}

\end{frame}

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\begin{frame}[c]

\frametitle{Some Simplification Rules}

\begin{center}

\bl{\begin{tabular}{rcl}

$r + \ZERO$  & $\equiv$ & $r$\\

$\ZERO + r$  & $\equiv$ & $r$\\

$r \cdot \ONE$ & $\equiv$ & $r$\\

$\ONE \cdot r$     & $\equiv$ & $r$\\

$r \cdot \ZERO$ & $\equiv$ & $\ZERO$\\

$\ZERO \cdot r$ & $\equiv$ & $\ZERO$\\

$r + r$ & $\equiv$ & $r$

\end{tabular}}

\end{center}

\end{frame}

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\begin{frame}[c]

\frametitle{The Specification for Matching}

\begin{bubble}[10cm]

\large

A regular expression \bl{$r$} matches a string~\bl{$s$} 

provided:

\begin{center}

\bl{$s \in L(r)$} 

\end{center}\medskip

\end{bubble}

\bigskip\bigskip

\ldots and the point of the this lecture is to decide this problem as

fast as possible (unlike Python, Ruby, Java etc)

\end{frame}

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\begin{frame}[c]

\frametitle{\begin{tabular}{c}\\[3cm]\huge\alert{Questions?}\end{tabular}}

\bigskip

homework (written exam 80\%)\\

coursework (20\%; first one today)\\

submission Fridays @ 18:00 -- accepted until Mondays

\mbox{}

\end{frame}

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\begin{frame}[t]

\frametitle{Semantic Derivative\\[5mm]}

\begin{itemize}

\item The \alert{\bf Semantic Derivative} of a 

\underline{language}\\ w.r.t.~to a character \bl{$c$}:

\bigskip

\begin{center}

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

\end{center}\bigskip

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

\begin{center}

\bl{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}

$\Der\,f\,A$ & $=$ & $\{\mathit{oo}, \mathit{rak}\}$\\

$\Der\,b\,A$ & $=$ &  $\{\mathit{ar}\}$\\  

$\Der\,a\,A$ & $=$ & $\{\}$\pause

\end{tabular}}

\end{center}\medskip

We can extend this definition to strings

\[

\bl{\Ders\,s\,A = \{s'\;|\;s\,@\,s' \in A\}}

\]

\end{itemize}

\end{frame}

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\begin{frame}[c]

\frametitle{\mbox{Brzozowski's Algorithm (1)}}