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% !TEX program = xelatex
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\documentclass[dvipsnames,14pt,t]{beamer}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../slides}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../graphics}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../langs}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{../data}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\hfuzz=220pt
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\lstset{language=Scala,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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style=mystyle,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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numbersep=0pt,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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numbers=none,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xleftmargin=0mm}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\pgfplotsset{compat=1.12}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% beamer stuff
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\slidecaption}{CFL 02, King's College London}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\LARGE Compilers and \\[-1mm]
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\LARGE Formal Languages (2)\\[3mm]
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\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\normalsize
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\begin{center}
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\begin{tabular}{ll}
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Email: & christian.urban at kcl.ac.uk\\
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Office Hours: & Thursdays 12 -- 14\\
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Location: & N7.07 (North Wing, Bush House)\\
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Slides \& Progs: & KEATS (also homework is there)\\
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\end{tabular}
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{
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Lets Implement an Efficient\\[-2mm]
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Regular Expression Matcher}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\footnotesize
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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{\normalsize Graphs: \bl{$a^{?\{n\}} \cdot a^{\{n\}}$} and strings \bl{$\underbrace{\,a\ldots a\,}_{n}$}}\medskip\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{@{}cc@{}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{axis}[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xlabel={$n$},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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x label style={at={(1.05,0.0)}},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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ylabel={time in secs},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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enlargelimits=false,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xtick={0,5,...,30},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xmax=33,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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ymax=35,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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ytick={0,5,...,30},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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scaled ticks=false,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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axis lines=left,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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width=5cm,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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height=4.5cm,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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legend entries={Python,Ruby},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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legend pos=north west,
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legend cell align=left]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{axis}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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&
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{axis}[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xlabel={$n$},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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x label style={at={(1.1,0.05)}},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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ylabel={time in secs},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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enlargelimits=false,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xtick={0,5000,...,10000},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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xmax=11500,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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ymax=35,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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ytick={0,5,...,30},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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scaled ticks=false,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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axis lines=left,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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width=5cm,
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height=4.5cm,
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legend entries={Scala V2,Scala V3},
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legend pos=north east,
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legend cell align=left]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{axis}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\small
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In the handouts is a similar graph for \bl{$(a^*)^* \cdot b$} and Java 8,
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JavaScript and Python.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\begin{frame}[c]
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%\frametitle{Evil Regular Expressions}
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%
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%\begin{itemize}
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%\item \alert{R}egular \alert{e}xpression \alert{D}enial \alert{o}f \alert{S}ervice (ReDoS)\bigskip
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%\item Evil regular expressions\medskip
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%\begin{itemize}
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%\item \bl{$a^{?\{n\}} \cdot a^{\{n\}}$}
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%\item \bl{$(a^*)^*$}
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%\item \bl{$([a$\,-\,$z]^+)^*$}
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%\item \bl{$(a + a \cdot a)^*$}
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%\item \bl{$(a + a^?)^*$}
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%\end{itemize}
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%
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%\item sometimes also called \alert{catastrophic backtracking}\bigskip
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%\item \small\ldots I hope you have watched the video by the
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% StackExchange engineer
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%\end{itemize}
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%
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[c]
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\frametitle{(Basic) Regular Expressions}
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Their inductive definition:
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\begin{center}
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\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
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\bl{$r$} & \bl{$::=$} & \bl{$\ZERO$} & nothing\\
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& \bl{$\mid$} & \bl{$\ONE$} & empty string / \pcode{""} / $[]$\\
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& \bl{$\mid$} & \bl{$c$} & character\\
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& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
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& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
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& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
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\end{tabular}
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\end{center}
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\vspace{\fill}
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Q: \;What about \;\bl{$r \cdot 0$} \; ?
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Languages (Sets of Strings)}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item A \alert{\bf Language} is a set of strings, for example\medskip
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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\bl{$\{[], hello, foobar, a, abc\}$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}\bigskip
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\item \alert{\bf Concatenation} for strings and languages
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{rcl}
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\bl{$foo\;@\;bar$} & \bl{$=$} & \bl{$foobar$}\medskip\\
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\bl{$A\;@\;B$} & \bl{$\dn$} & \bl{$\{ s_1\,@\,s_2 \;\mid\; s_1 \in A \wedge s_2 \in B\}$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bigskip
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\small
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item [] For example \bl{$A = \{foo, bar\}$}, \bl{$B = \{a, b\}$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{A \,@\, B = \{fooa, foob, bara, barb\}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Two Corner Cases}
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\Large
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\begin{center}
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\bl{$A \,@\, \{[]\} = \;?$}\bigskip\bigskip\pause\\
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\bl{$A \,@\, \{\} = \;?$}
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\end{center}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{
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The Meaning of a\\[-2mm]
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Regular Expression}
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...all the strings a regular expression can match.
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\begin{center}
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\begin{tabular}{rcl}
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\bl{$L(\ZERO)$} & \bl{$\dn$} & \bl{$\{\}$}\\
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\bl{$L(\ONE)$} & \bl{$\dn$} & \bl{$\{[]\}$}\\
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\bl{$L(c)$} & \bl{$\dn$} & \bl{$\{[c]\}$}\\
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\bl{$L(r_1 + r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \cup L(r_2)$}\\
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\bl{$L(r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\
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\bl{$L(r^*)$} & \bl{$\dn$} & \\
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\end{tabular}
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\end{center}
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\begin{textblock}{14}(1.5,13.5)\small
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\bl{$L$} is a function from regular expressions to
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sets of strings (languages):\smallskip\\
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\bl{\quad$L$ : Rexp $\Rightarrow$ Set$[$String$]$}
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\end{textblock}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{The Power Operation}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{itemize}
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\item The \alert{\textbf{\boldmath$n$th Power}} of a language:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tabular}{lcl}
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\bl{$A^0$} & \bl{$\dn$} & \bl{$\{[]\}$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$A^{n+1}$} & \bl{$\dn$} & \bl{$A \,@\, A^n$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}\bigskip
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item[] For example
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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\begin{tabular}{lcl@{\hspace{10mm}}l}
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\bl{$A^4$} & \bl{$=$} & \bl{$A \,@\, A \,@\, A \,@\, A$} & \bl{$(@\,\{[]\})$}\\
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\bl{$A^1$} & \bl{$=$} & \bl{$A$} & \bl{$(@\,\{[]\})$}\\
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\bl{$A^0$} & \bl{$=$} & \bl{$\{[]\}$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
255 |
|
261
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
256 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
257 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
258 |
\end{frame}
|
16
|
259 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
260 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
261 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
262 |
\begin{frame}[c]
|
638
|
263 |
\frametitle{Homework Question}
|
|
264 |
|
|
265 |
\begin{itemize}
|
|
266 |
\item Say \bl{$A = \{[a],[b],[c],[d]\}$}.\bigskip
|
|
267 |
|
|
268 |
\item[]
|
|
269 |
How many strings are in \bl{$A^4$}\,?
|
|
270 |
\bigskip\medskip\pause
|
|
271 |
|
|
272 |
|
|
273 |
\item[]
|
|
274 |
What if \bl{$A = \{[a],[b],[c],[]\}$};\\
|
|
275 |
how many strings are then in \bl{$A^4$}\,?
|
|
276 |
\end{itemize}
|
|
277 |
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
278 |
\end{frame}
|
638
|
279 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
280 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
281 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
282 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
283 |
\frametitle{The Star Operation}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
284 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
285 |
\begin{itemize}
|
514
|
286 |
\item The \alert{\bf Kleene Star} of a \underline{language}:
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
287 |
\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
288 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
289 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
290 |
\begin{tabular}{c}
|
431
|
291 |
\bl{$A\star \dn \bigcup_{0\le n} A^n$}
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
292 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
293 |
\end{center}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
294 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
295 |
\item[] This expands to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
296 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
297 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
298 |
\bl{A^0 \cup A^1 \cup A^2 \cup A^3 \cup A^4 \cup \ldots}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
299 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
300 |
|
565
|
301 |
or
|
|
302 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
303 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
304 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
305 |
\bl{\{[]\} \;\cup\; A \;\cup\; A\,@\,A \;\cup\;
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
306 |
A\,@\,A\,@\,A \;\cup\; A\,@\,A\,@\,A\,@\,A \cup \ldots}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
307 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
308 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
309 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
310 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
311 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
312 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
313 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
314 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
315 |
\begin{frame}[c]
|
565
|
316 |
\frametitle{
|
|
317 |
The Meaning of a\\[-2mm]
|
|
318 |
Regular Expression}
|
507
|
319 |
|
638
|
320 |
...all the strings a regular expression can match.
|
|
321 |
|
|
322 |
\begin{center}
|
507
|
323 |
\begin{tabular}{rcl}
|
|
324 |
\bl{$L(\ZERO)$} & \bl{$\dn$} & \bl{$\{\}$}\\
|
|
325 |
\bl{$L(\ONE)$} & \bl{$\dn$} & \bl{$\{[]\}$}\\
|
|
326 |
\bl{$L(c)$} & \bl{$\dn$} & \bl{$\{[c]\}$}\\
|
|
327 |
\bl{$L(r_1 + r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \cup L(r_2)$}\\
|
638
|
328 |
\bl{$L(r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\
|
507
|
329 |
\bl{$L(r^*)$} & \bl{$\dn$} & \bl{$(L(r))\star \quad\dn \bigcup_{0 \le n} L(r)^n$}\\
|
|
330 |
\end{tabular}
|
638
|
331 |
\end{center}
|
507
|
332 |
|
638
|
333 |
%\begin{textblock}{9}(6,12)\small
|
|
334 |
%\bl{$L$} is a function from regular expressions to
|
|
335 |
%sets of strings (languages):\smallskip\\
|
|
336 |
%\bl{$L$ : Rexp $\Rightarrow$ Set$[$String$]$}
|
|
337 |
%\end{textblock}
|
507
|
338 |
|
|
339 |
\end{frame}
|
|
340 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
341 |
|
638
|
342 |
|
|
343 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
344 |
\begin{frame}[c]
|
|
345 |
\frametitle{
|
|
346 |
When Are Two Regular\\[-1mm]
|
|
347 |
Expressions Equivalent?}
|
|
348 |
|
|
349 |
\begin{bubble}[10cm]
|
|
350 |
\large
|
|
351 |
Two regular expressions \bl{$r_1$} and \bl{$r_2$} are equivalent
|
|
352 |
provided:\medskip
|
|
353 |
\begin{center}
|
|
354 |
\bl{$r_1 \equiv r_2 \;\;\dn\;\; L(r_1) = L(r_2)$}
|
|
355 |
\end{center}\medskip
|
|
356 |
\end{bubble}
|
|
357 |
|
|
358 |
\end{frame}
|
|
359 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
360 |
|
|
361 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
362 |
\begin{frame}[c]
|
|
363 |
\frametitle{Some Concrete Equivalences}
|
|
364 |
|
|
365 |
\begin{center}
|
|
366 |
\bl{\begin{tabular}{rcl}
|
|
367 |
$(a + b) + c$ & $\equiv$ & $a + (b + c)$\\
|
|
368 |
$a + a$ & $\equiv$ & $a$\\
|
|
369 |
$a + b$ & $\equiv$ & $b + a$\\
|
|
370 |
$(a \cdot b) \cdot c$ & $\equiv$ & $a \cdot (b \cdot c)$\\
|
|
371 |
$c \cdot (a + b)$ & $\equiv$ & $(c \cdot a) + (c \cdot b)$\bigskip\bigskip\\\pause
|
|
372 |
$a \cdot a$ & $\not\equiv$ & $a$\\
|
|
373 |
$a + (b \cdot c)$ & $\not\equiv$ & $(a + b) \cdot (a + c)$\\
|
|
374 |
\end{tabular}}
|
|
375 |
\end{center}
|
|
376 |
|
|
377 |
\end{frame}
|
|
378 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
379 |
|
|
380 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
381 |
\begin{frame}[c]
|
|
382 |
\frametitle{Some Corner Cases}
|
|
383 |
|
|
384 |
\begin{center}
|
|
385 |
\bl{\begin{tabular}{rcl}
|
|
386 |
$a \cdot \ZERO$ & $\not\equiv$ & $a$\\
|
|
387 |
$a + \ONE$ & $\not\equiv$ & $a$\\
|
|
388 |
$\ONE$ & $\equiv$ & $\ZERO^*$\\
|
|
389 |
$\ONE^*$ & $\equiv$ & $\ONE$\\
|
|
390 |
$\ZERO^*$ & $\not\equiv$ & $\ZERO$\\
|
|
391 |
\end{tabular}}
|
|
392 |
\end{center}
|
|
393 |
|
|
394 |
\end{frame}
|
|
395 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
396 |
|
|
397 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
398 |
\begin{frame}[c]
|
|
399 |
\frametitle{Some Simplification Rules}
|
|
400 |
|
|
401 |
\begin{center}
|
|
402 |
\bl{\begin{tabular}{rcl}
|
|
403 |
$r + \ZERO$ & $\equiv$ & $r$\\
|
|
404 |
$\ZERO + r$ & $\equiv$ & $r$\\
|
|
405 |
$r \cdot \ONE$ & $\equiv$ & $r$\\
|
|
406 |
$\ONE \cdot r$ & $\equiv$ & $r$\\
|
|
407 |
$r \cdot \ZERO$ & $\equiv$ & $\ZERO$\\
|
|
408 |
$\ZERO \cdot r$ & $\equiv$ & $\ZERO$\\
|
|
409 |
$r + r$ & $\equiv$ & $r$
|
|
410 |
\end{tabular}}
|
|
411 |
\end{center}
|
|
412 |
|
|
413 |
\end{frame}
|
|
414 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
415 |
|
|
416 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
417 |
\begin{frame}[c]
|
|
418 |
\frametitle{The Specification for Matching}
|
|
419 |
|
|
420 |
\begin{bubble}[10cm]
|
|
421 |
\large
|
|
422 |
A regular expression \bl{$r$} matches a string~\bl{$s$}
|
|
423 |
provided:
|
|
424 |
\begin{center}
|
|
425 |
\bl{$s \in L(r)$}
|
|
426 |
\end{center}\medskip
|
|
427 |
\end{bubble}
|
|
428 |
|
|
429 |
\bigskip\bigskip
|
|
430 |
|
|
431 |
\ldots and the point of the this lecture is to decide this problem as
|
|
432 |
fast as possible (unlike Python, Ruby, Java etc)
|
|
433 |
|
|
434 |
\end{frame}
|
|
435 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
436 |
|
|
437 |
|
566
|
438 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
439 |
\begin{frame}[c]
|
|
440 |
\frametitle{\begin{tabular}{c}\\[3cm]\huge\alert{Questions?}\end{tabular}}
|
|
441 |
|
|
442 |
\bigskip
|
|
443 |
homework (written exam 80\%)\\
|
|
444 |
coursework (20\%; first one today)\\
|
567
|
445 |
submission Fridays @ 18:00 -- accepted until Mondays
|
566
|
446 |
\mbox{}
|
|
447 |
\end{frame}
|
|
448 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
449 |
|
|
450 |
|
507
|
451 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
565
|
452 |
\begin{frame}[t]
|
|
453 |
\frametitle{Semantic Derivative\\[5mm]}
|
|
454 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
455 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
456 |
\item The \alert{\bf Semantic Derivative} of a
|
565
|
457 |
\underline{language}\\ w.r.t.~to a character \bl{$c$}:
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
458 |
\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
459 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
460 |
\begin{center}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
461 |
\bl{$\Der\,c\,A \dn \{ s \;|\; c\!::\!s \in A\}$ }
|
565
|
462 |
\end{center}\bigskip
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
463 |
|
630
|
464 |
For \bl{$A = \{\mathit{foo}, \mathit{bar}, \mathit{frak}\}$} then
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
465 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
466 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
467 |
\bl{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}
|
630
|
468 |
$\Der\,f\,A$ & $=$ & $\{\mathit{oo}, \mathit{rak}\}$\\
|
|
469 |
$\Der\,b\,A$ & $=$ & $\{\mathit{ar}\}$\\
|
565
|
470 |
$\Der\,a\,A$ & $=$ & $\{\}$\pause
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
471 |
\end{tabular}}
|
638
|
472 |
\end{center}\medskip
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
473 |
|
630
|
474 |
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
475 |
We can extend this definition to strings
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
476 |
\[
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
477 |
\bl{\Ders\,s\,A = \{s'\;|\;s\,@\,s' \in A\}}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
478 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
479 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
480 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
481 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
482 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
483 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
484 |
|
512
|
485 |
|
|
486 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
487 |
\begin{frame}[c]
|
573
|
488 |
\frametitle{\mbox{Brzozowski's Algorithm (1)}}
|
11
|
489 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
490 |
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
491 |
\ldots{}whether a regular expression can match the empty string:
|
11
|
492 |
\begin{center}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
493 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
429
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
494 |
\bl{$nullable(\ZERO)$} & \bl{$\dn$} & \bl{\textit{false}}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
495 |
\bl{$nullable(\ONE)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
496 |
\bl{$nullable (c)$} & \bl{$\dn$} & \bl{\textit{false}}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
497 |
\bl{$nullable (r_1 + r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \vee nullable(r_2)$} \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
498 |
\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$} \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
499 |
\bl{$nullable (r^*)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
500 |
\end{tabular}
|
16
|
501 |
\end{center}
|
|
502 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
503 |
\end{frame}
|
11
|
504 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
505 |
|
|
506 |
|
|
507 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
508 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
509 |
\frametitle{The Derivative of a Rexp}
|
7
|
510 |
|
|
511 |
\large
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
512 |
If \bl{$r$} matches the string \bl{$c\!::\!s$}, what is a regular
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
513 |
expression that matches just \bl{$s$}?\bigskip\bigskip\bigskip\bigskip
|
7
|
514 |
|
|
515 |
\small
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
516 |
\bl{$\der\,c\,r$} gives the answer, Brzozowski 1964
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
517 |
\end{frame}
|
7
|
518 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
519 |
|
|
520 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
521 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
522 |
\frametitle{The Derivative of a Rexp}
|
7
|
523 |
|
|
524 |
\begin{center}
|
|
525 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
526 |
\bl{$\der\, c\, (\ZERO)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
527 |
\bl{$\der\, c\, (\ONE)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
528 |
\bl{$\der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
529 |
\bl{$\der\, c\, (r_1 + r_2)$} & \bl{$\dn$} & \bl{$\der\, c\, r_1 + \der\, c\, r_2$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
530 |
\bl{$\der\, c\, (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{if $nullable (r_1)$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
531 |
& & \bl{then $(\der\,c\,r_1) \cdot r_2 + \der\, c\, r_2$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
532 |
& & \bl{else $(\der\, c\, r_1) \cdot r_2$}\\
|
638
|
533 |
\bl{$\der\, c\, (r^*)$} & \bl{$\dn$} & \bl{$(\der\,c\,r) \cdot (r^*)$} &\medskip\bigskip\\\pause
|
7
|
534 |
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
535 |
\bl{$\ders\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
536 |
\bl{$\ders\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\ders\,s\,(\der\,c\,r)$} & \\
|
7
|
537 |
\end{tabular}
|
|
538 |
\end{center}
|
|
539 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
540 |
\end{frame}
|
7
|
541 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
542 |
|
|
543 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
544 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
545 |
\frametitle{Examples}
|
7
|
546 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
547 |
Given \bl{$r \dn ((a \cdot b) + b)^*$} what is
|
7
|
548 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
549 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
550 |
\begin{tabular}{l}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
551 |
\bl{$\der\,a\,r =\;?$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
552 |
\bl{$\der\,b\,r =\;?$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
553 |
\bl{$\der\,c\,r =\;?$}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
554 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
555 |
\end{center}
|
7
|
556 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
557 |
\end{frame}
|
7
|
558 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
559 |
|
17
|
560 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
431
|
561 |
\begin{frame}[c]
|
565
|
562 |
\frametitle{\mbox{The Brzozowski Algorithm}}
|
431
|
563 |
|
|
564 |
\begin{center}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
565 |
\begin{tabular}{l}
|
565
|
566 |
\bl{$\textit{matches}\,r\,s \dn \textit{nullable}(\ders\;s\;r)$}
|
431
|
567 |
\end{tabular}
|
|
568 |
\end{center}
|
|
569 |
|
|
570 |
|
|
571 |
\end{frame}
|
|
572 |
%%%%%%%%%%%%%%%%%
|
|
573 |
|
|
574 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
17
|
575 |
\begin{frame}[t]
|
565
|
576 |
\frametitle{Brzozowski: An Example}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
577 |
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
578 |
Does \bl{$r_1$} match \bl{$abc$}?
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
579 |
\begin{center}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
580 |
\begin{tabular}{@{}rl@{}l@{}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
581 |
Step 1: & build derivative of \bl{$a$} and \bl{$r_1$} & \bl{$(r_2 = \der\,a\,r_1)$}\smallskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
582 |
Step 2: & build derivative of \bl{$b$} and \bl{$r_2$} & \bl{$(r_3 = \der\,b\,r_2)$}\smallskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
583 |
Step 3: & build derivative of \bl{$c$} and \bl{$r_3$} & \bl{$(r_4 = \der\,c\,r_3)$}\smallskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
584 |
Step 4: & the string is exhausted: & \bl{($nullable(r_4))$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
585 |
& test whether \bl{$r_4$} can recognise\\
|
565
|
586 |
& the empty string\medskip\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
587 |
Output: & result of the test\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
588 |
& $\Rightarrow \bl{\textit{true}} \,\text{or}\,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
589 |
\bl{\textit{false}}$\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
590 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
591 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
592 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
593 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
594 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
595 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
596 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
597 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
598 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
599 |
\frametitle{The Idea of the Algorithm}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
600 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
601 |
If we want to recognise the string \bl{$abc$} with regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
602 |
expression \bl{$r_1$} then\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
603 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
604 |
\begin{enumerate}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
605 |
\item \bl{$\Der\,a\,(L(r_1))$}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
606 |
\item \bl{$\Der\,b\,(\Der\,a\,(L(r_1)))$}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
607 |
\item \bl{$\Der\,c\,(\Der\,b\,(\Der\,a\,(L(r_1))))$}\bigskip
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
608 |
\item finally we test whether the empty string is in this
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
609 |
set; same for \bl{$\Ders\,abc\,(L(r_1))$}.\medskip
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
610 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
611 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
612 |
The matching algorithm works similarly, just over regular expressions instead of sets.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
613 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
614 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
615 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
616 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
617 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
618 |
\begin{frame}[c]
|
638
|
619 |
\frametitle{Oops\ldots \boldmath\;$a^{?\{n\}} \cdot a^{\{n\}}$}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
620 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
621 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
622 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
623 |
\begin{axis}[
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
624 |
xlabel={$n$},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
625 |
x label style={at={(1.05,0.0)}},
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
626 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
627 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
628 |
xtick={0,5,...,30},
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
629 |
xmax=31,
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
630 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
631 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
632 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
633 |
width=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
634 |
height=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
635 |
legend entries={Python,Ruby,Scala V1},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
636 |
legend pos=outer north east,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
637 |
legend cell align=left
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
638 |
]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
639 |
\addplot[blue,mark=*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
640 |
table {re-python.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
641 |
\addplot[brown,mark=pentagon*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
642 |
table {re-ruby.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
643 |
\addplot[red,mark=triangle*,mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
644 |
table {re1.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
645 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
646 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
647 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
648 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
649 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
650 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
651 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
652 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
565
|
653 |
\begin{frame}[t]
|
|
654 |
\frametitle{A Problem\medskip}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
655 |
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
656 |
We represented the ``n-times'' \bl{$a^{\{n\}}$} as a
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
657 |
sequence regular expression:
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
658 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
659 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
660 |
\begin{tabular}{rl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
661 |
1: & \bl{$a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
662 |
2: & \bl{$a\cdot a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
663 |
3: & \bl{$a\cdot a\cdot a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
664 |
& \ldots\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
665 |
13: & \bl{$a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
666 |
& \ldots\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
667 |
20:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
668 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
669 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
670 |
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
671 |
This problem is aggravated with \bl{$a^?$} being represented
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
672 |
as \bl{$a + \ONE$}.
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
673 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
674 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
675 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
676 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
677 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
678 |
\frametitle{Solving the Problem}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
679 |
|
514
|
680 |
What happens if we extend our regular expressions with explicit
|
|
681 |
constructors
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
682 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
683 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
684 |
\begin{tabular}{rcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
685 |
\bl{$r$} & \bl{$::=$} & \bl{\ldots}\\
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
686 |
& \bl{$\mid$} & \bl{$r^{\{n\}}$}\\
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
687 |
& \bl{$\mid$} & \bl{$r^?$}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
688 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
689 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
690 |
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
691 |
What is their meaning?\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
692 |
What are the cases for \bl{$nullable$} and \bl{$\der$}?
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
693 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
694 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
695 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
696 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
697 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
698 |
\begin{frame}[t]
|
638
|
699 |
\frametitle{Brzozowski: \boldmath$a^{?\{n\}} \cdot a^{\{n\}}$}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
700 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
701 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
702 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
703 |
\begin{axis}[
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
704 |
xlabel={$n$},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
705 |
x label style={at={(1.05,0.0)}},
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
706 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
707 |
enlargelimits=false,
|
434
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
708 |
xtick={0,200,...,1100},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
709 |
xmax=1200,
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
710 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
711 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
712 |
axis lines=left,
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
713 |
width=8cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
714 |
height=6.5cm,
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
715 |
legend entries={Python,Ruby,Scala V1,Scala V2},
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
716 |
legend pos=outer north east,
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
717 |
legend cell align=left
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
718 |
]
|
434
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
719 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
720 |
\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
721 |
\addplot[red,mark=triangle*,mark options={fill=white}] table {re1.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
722 |
\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
723 |
\end{axis}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
724 |
\end{tikzpicture}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
725 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
726 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
727 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
728 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
729 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
730 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
731 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
732 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
733 |
\frametitle{Examples}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
734 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
735 |
Recall the example of \bl{$r \dn ((a \cdot b) + b)^*$} with
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
736 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
737 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
738 |
\begin{tabular}{l}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
739 |
\bl{$\der\,a\,r = ((\ONE \cdot b) + \ZERO) \cdot r$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
740 |
\bl{$\der\,b\,r = ((\ZERO \cdot b) + \ONE)\cdot r$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
741 |
\bl{$\der\,c\,r = ((\ZERO \cdot b) + \ZERO)\cdot r$}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
742 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
743 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
744 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
745 |
What are these regular expressions equivalent to?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
746 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
747 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
748 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
749 |
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
750 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
751 |
\begin{frame}[c]
|
565
|
752 |
\frametitle{Simplification Rules}
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
753 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
754 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
755 |
\bl{\begin{tabular}{rcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
756 |
$r + \ZERO$ & $\Rightarrow$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
757 |
$\ZERO + r$ & $\Rightarrow$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
758 |
$r \cdot \ONE$ & $\Rightarrow$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
759 |
$\ONE \cdot r$ & $\Rightarrow$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
760 |
$r \cdot \ZERO$ & $\Rightarrow$ & $\ZERO$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
761 |
$\ZERO \cdot r$ & $\Rightarrow$ & $\ZERO$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
762 |
$r + r$ & $\Rightarrow$ & $r$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
763 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
764 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
765 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
766 |
\footnotesize
|
565
|
767 |
\lstinputlisting[firstline=22,lastline=26]{../progs/app6.scala}
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
768 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
769 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
770 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
771 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
772 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
773 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
774 |
\footnotesize
|
565
|
775 |
\lstinputlisting[firstline=1,lastline=21]{../progs/app6.scala}
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
776 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
777 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
778 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
779 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
780 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
781 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
782 |
\begin{frame}[t]
|
638
|
783 |
\frametitle{Brzozowski: \boldmath$a^{?\{n\}} \cdot a^{\{n\}}$}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
784 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
785 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
786 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
787 |
\begin{axis}[
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
788 |
xlabel={$n$},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
789 |
x label style={at={(1.05,0.0)}},
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
790 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
791 |
enlargelimits=false,
|
442
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
792 |
xtick={0,5000,...,10000},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
793 |
xmax=11000,
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
794 |
ymax=35,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
795 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
796 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
797 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
798 |
width=9cm,
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
799 |
height=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
800 |
legend entries={Scala V2,Scala V3},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
801 |
legend pos=north east
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
802 |
]
|
434
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
803 |
\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
804 |
\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
805 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
806 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
807 |
\end{center}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
808 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
809 |
\end{frame}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
810 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
811 |
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
812 |
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
813 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
814 |
\begin{frame}[c]
|
565
|
815 |
\frametitle{
|
638
|
816 |
Another Example\\[-2mm]
|
|
817 |
in Java 8, Python and JavaScript}
|
565
|
818 |
|
638
|
819 |
\bigskip
|
565
|
820 |
\begin{center}
|
|
821 |
\begin{tikzpicture}
|
|
822 |
\begin{axis}[
|
|
823 |
xlabel={$n$},
|
|
824 |
x label style={at={(1.05,0.0)}},
|
|
825 |
ylabel={time in secs},
|
|
826 |
enlargelimits=false,
|
|
827 |
xtick={0,5,...,30},
|
|
828 |
xmax=33,
|
|
829 |
ymax=35,
|
|
830 |
ytick={0,10,...,30},
|
|
831 |
scaled ticks=false,
|
|
832 |
axis lines=left,
|
|
833 |
width=9cm,
|
638
|
834 |
height=5.5cm,
|
|
835 |
legend entries={Java 8, Python, JavaScript},
|
565
|
836 |
legend pos=north west,
|
|
837 |
legend cell align=left]
|
|
838 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
|
|
839 |
\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
|
638
|
840 |
\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
|
565
|
841 |
\end{axis}
|
|
842 |
\end{tikzpicture}
|
|
843 |
\end{center}
|
|
844 |
|
|
845 |
Regex: \bl{$(a^*)^* \cdot b$}
|
|
846 |
|
|
847 |
Strings of the form \bl{$\underbrace{\,a\ldots a\,}_{n}$}
|
|
848 |
|
|
849 |
\end{frame}
|
|
850 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
851 |
|
|
852 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
853 |
\begin{frame}[c]
|
638
|
854 |
\frametitle{Same Example in Java 9+}
|
565
|
855 |
|
|
856 |
\begin{center}
|
|
857 |
\begin{tikzpicture}
|
|
858 |
\begin{axis}[
|
|
859 |
xlabel={$n$},
|
|
860 |
x label style={at={(1.09,-0.15)}},
|
|
861 |
ylabel={time in secs},
|
|
862 |
scaled x ticks=false,
|
|
863 |
enlargelimits=false,
|
|
864 |
xtick distance=10000,
|
|
865 |
xmax=44000,
|
|
866 |
ytick={0,10,...,30},
|
|
867 |
ymax=35,
|
|
868 |
axis lines=left,
|
|
869 |
width=9cm,
|
|
870 |
height=5cm,
|
|
871 |
legend entries={Java \liningnums{9}+},
|
|
872 |
legend pos=north west,
|
|
873 |
legend cell align=left]
|
|
874 |
\addplot[blue,mark=square*,mark options={fill=white}] table {re-java9.data};
|
|
875 |
\end{axis}
|
|
876 |
\end{tikzpicture}
|
|
877 |
\end{center}
|
|
878 |
|
|
879 |
Regex: \bl{$(a^*)^* \cdot b$}
|
|
880 |
|
|
881 |
Strings of the form \bl{$\underbrace{\,a\ldots a\,}_{n}$}
|
|
882 |
|
|
883 |
\end{frame}
|
|
884 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
885 |
|
|
886 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
887 |
\begin{frame}[c]
|
646
|
888 |
\frametitle{\ldots{}and with Brzozowski}
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
889 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
890 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
891 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
892 |
\begin{axis}[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
893 |
xlabel={$n$},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
894 |
x label style={at={(1.09,0.0)}},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
895 |
ylabel={time in secs},
|
565
|
896 |
scaled x ticks=false,
|
|
897 |
xtick distance=2000000,
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
898 |
enlargelimits=false,
|
565
|
899 |
xmax=6400000,
|
514
|
900 |
ytick={0,10,...,30},
|
|
901 |
ymax=35,
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
902 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
903 |
width=9cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
904 |
height=5cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
905 |
legend entries={Scala V3},
|
565
|
906 |
legend pos=north west,
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
907 |
legend cell align=left]
|
514
|
908 |
%\addplot[green,mark=square*,mark options={fill=white}] table {re2a.data};
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
909 |
\addplot[black,mark=square*,mark options={fill=white}] table {re3a.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
910 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
911 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
912 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
913 |
|
565
|
914 |
Regex: \bl{$(a^*)^* \cdot b$}
|
|
915 |
|
|
916 |
Strings of the form \bl{$\underbrace{\,a\ldots a\,}_{n}$}
|
|
917 |
|
435
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
918 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
919 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
920 |
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
921 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
573
|
922 |
\begin{frame}[c]
|
|
923 |
\frametitle{\mbox{What is good about this Alg.}}
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
924 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
925 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
926 |
\item extends to most regular expressions, for example
|
573
|
927 |
\bl{$\sim r$} (next slide)
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
928 |
|
573
|
929 |
\item is easy to implement in a functional language (slide after)
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
930 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
931 |
\item the algorithm is already quite old; there is still
|
514
|
932 |
work to be done to use it as a tokenizer (that is relatively new work)
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
933 |
|
638
|
934 |
\item we can prove its correctness\ldots (several slides later)
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
935 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
936 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
937 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
938 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
939 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
940 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
566
|
941 |
\begin{frame}[c]
|
|
942 |
\frametitle{Negation of Regular Expr's}
|
|
943 |
|
|
944 |
\begin{itemize}
|
|
945 |
\item \bl{$\sim{}r$} \hspace{6mm} (everything that \bl{$r$} cannot recognise)\medskip
|
|
946 |
\item \bl{$L(\sim{}r) \dn UNIV - L(r)$}\medskip
|
|
947 |
\item \bl{$nullable (\sim{}r) \dn not\, (nullable(r))$}\medskip
|
|
948 |
\item \bl{$der\,c\,(\sim{}r) \dn \;\sim{}(der\,c\,r)$}
|
|
949 |
\end{itemize}\bigskip\pause
|
|
950 |
|
|
951 |
Used often for recognising comments:
|
|
952 |
|
|
953 |
\[
|
|
954 |
\bl{/ \cdot * \cdot (\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot * \cdot /}
|
|
955 |
\]
|
|
956 |
|
|
957 |
\end{frame}
|
|
958 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
959 |
|
|
960 |
|
|
961 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
962 |
\begin{frame}[c]
|
|
963 |
\frametitle{Coursework}
|
|
964 |
|
|
965 |
\underline{Strand 1:}
|
|
966 |
|
|
967 |
\begin{itemize}
|
638
|
968 |
\item Submission on Friday 11 October\\accepted until Monday 14 @ 18:00\medskip
|
566
|
969 |
\item source code needs to be submitted as well\medskip
|
|
970 |
\item you can re-use my Scala code from KEATS \\
|
|
971 |
or use any programming language you like\medskip
|
|
972 |
\item \small https://nms.kcl.ac.uk/christian.urban/ProgInScala2ed.pdf\normalsize
|
|
973 |
\end{itemize}
|
|
974 |
|
|
975 |
\end{frame}
|
|
976 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
977 |
|
|
978 |
|
|
979 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
980 |
\begin{frame}[t]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
981 |
\frametitle{Proofs about Rexps}
|
17
|
982 |
|
|
983 |
Remember their inductive definition:\\[5cm]
|
|
984 |
|
|
985 |
\begin{textblock}{6}(5,5)
|
|
986 |
\begin{tabular}{@ {}rrl}
|
429
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
987 |
\bl{$r$} & \bl{$::=$} & \bl{$\ZERO$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
988 |
& \bl{$\mid$} & \bl{$\ONE$} \\
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
989 |
& \bl{$\mid$} & \bl{$c$} \\
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
990 |
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$}\\
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
991 |
& \bl{$\mid$} & \bl{$r_1 + r_2$} \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
992 |
& \bl{$\mid$} & \bl{$r^*$} \\
|
17
|
993 |
\end{tabular}
|
|
994 |
\end{textblock}
|
|
995 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
996 |
If we want to prove something, say a property \bl{$P(r)$}, for all regular expressions \bl{$r$} then \ldots
|
17
|
997 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
998 |
\end{frame}
|
17
|
999 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1000 |
|
|
1001 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1002 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1003 |
\frametitle{Proofs about Rexp (2)}
|
17
|
1004 |
|
|
1005 |
\begin{itemize}
|
429
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1006 |
\item \bl{$P$} holds for \bl{$\ZERO$}, \bl{$\ONE$} and \bl{c}\bigskip
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1007 |
\item \bl{$P$} holds for \bl{$r_1 + r_2$} under the assumption that \bl{$P$} already
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1008 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1009 |
\item \bl{$P$} holds for \bl{$r_1 \cdot r_2$} under the assumption that \bl{$P$} already
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1010 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1011 |
\item \bl{$P$} holds for \bl{$r^*$} under the assumption that \bl{$P$} already
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1012 |
holds for \bl{$r$}.
|
17
|
1013 |
\end{itemize}
|
|
1014 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1015 |
\end{frame}
|
17
|
1016 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1017 |
|
|
1018 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1019 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1020 |
\frametitle{Proofs about Rexp (3)}
|
17
|
1021 |
|
|
1022 |
Assume \bl{$P(r)$} is the property:
|
|
1023 |
|
|
1024 |
\begin{center}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1025 |
\bl{$nullable(r)$} if and only if \bl{$[] \in L(r)$}
|
17
|
1026 |
\end{center}
|
|
1027 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1028 |
\end{frame}
|
17
|
1029 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1030 |
|
|
1031 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1032 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1033 |
\frametitle{Proofs about Rexp (4)}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1034 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1035 |
\begin{center}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1036 |
\bl{\begin{tabular}{r@{\hspace{2mm}}c@{\hspace{2mm}}l}
|
429
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1037 |
$rev(\ZERO)$ & $\dn$ & $\ZERO$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1038 |
$rev(\ONE)$ & $\dn$ & $\ONE$\\
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1039 |
$rev(c)$ & $\dn$ & $c$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1040 |
$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1041 |
$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1042 |
$rev(r^*)$ & $\dn$ & $rev(r)^*$\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1043 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1044 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1045 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1046 |
We can prove
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1047 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1048 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1049 |
\bl{$L(rev(r)) = \{s^{-1} \;|\; s \in L(r)\}$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1050 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1051 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1052 |
by induction on \bl{$r$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1053 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1054 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1055 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1056 |
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1057 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1058 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1059 |
\begin{frame}[c]
|
638
|
1060 |
\frametitle{Correctness Proof \\[-1mm]
|
|
1061 |
for our Matcher}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1062 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1063 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1064 |
\item We started from
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1065 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1066 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1067 |
\begin{tabular}{rp{4cm}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1068 |
& \bl{$s \in L(r)$}\medskip\\
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1069 |
\bl{$\Leftrightarrow$} & \bl{$[] \in \Ders\,s\,(L(r))$}\pause
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1070 |
\end{tabular}
|
638
|
1071 |
\end{center}\bigskip
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1072 |
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1073 |
\item if we can show \bl{$\Ders\, s\,(L(r)) = L(\ders\,s\,r)$} we
|
638
|
1074 |
have\bigskip
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1075 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1076 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1077 |
\begin{tabular}{rp{4cm}}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1078 |
\bl{$\Leftrightarrow$} & \bl{$[] \in L(\ders\,s\,r)$}\medskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1079 |
\bl{$\Leftrightarrow$} & \bl{$nullable(\ders\,s\,r)$}\medskip\\
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1080 |
\bl{$\dn$} & \bl{$matches\,s\,r$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1081 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1082 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1083 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1084 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1085 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1086 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1087 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1088 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1089 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1090 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1091 |
\frametitle{Proofs about Rexp (5)}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1092 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1093 |
Let \bl{$Der\,c\,A$} be the set defined as
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1094 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1095 |
\begin{center}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1096 |
\bl{$\Der\,c\,A \dn \{ s \;|\; c\!::\!s \in A\}$ }
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1097 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1098 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1099 |
We can prove
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1100 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1101 |
\begin{center}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1102 |
\bl{$L(\der\,c\,r) = \Der\,c\,(L(r))$}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1103 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1104 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1105 |
by induction on \bl{$r$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1106 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1107 |
\end{frame}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1108 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1109 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1110 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1111 |
\begin{frame}[c]
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1112 |
\frametitle{Proofs about Strings}
|
17
|
1113 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1114 |
If we want to prove something, say a property \bl{$P(s)$}, for all strings \bl{$s$} then \ldots\bigskip
|
17
|
1115 |
|
|
1116 |
\begin{itemize}
|
|
1117 |
\item \bl{$P$} holds for the empty string, and\medskip
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1118 |
\item \bl{$P$} holds for the string \bl{$c\!::\!s$} under the assumption that \bl{$P$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1119 |
already holds for \bl{$s$}
|
17
|
1120 |
\end{itemize}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1121 |
\end{frame}
|
17
|
1122 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1123 |
|
19
|
1124 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1125 |
\begin{frame}[c]
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1126 |
\frametitle{Proofs about Strings (2)}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1127 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1128 |
We can then prove
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1129 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1130 |
\begin{center}
|
433
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1131 |
\bl{$\Ders\,s\,(L(r)) = L(\ders\,s\,r)$}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1132 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1133 |
|
19
|
1134 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1135 |
We can finally prove
|
19
|
1136 |
|
|
1137 |
\begin{center}
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1138 |
\bl{$matches\,s\,r$} if and only if \bl{$s \in L(r)$}
|
19
|
1139 |
\end{center}
|
|
1140 |
|
|
1141 |
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1142 |
\end{frame}
|
19
|
1143 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
17
|
1144 |
|
436
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1145 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1146 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1147 |
\frametitle{Epilogue}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1148 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1149 |
\footnotesize
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1150 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1151 |
\begin{tabular}{cc}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1152 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1153 |
\begin{axis}[
|
442
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1154 |
title={\small{}Graph: \bl{$a^{?\{n\}} \cdot a^{\{n\}}$}},
|
436
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1155 |
xlabel={$n$},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1156 |
x label style={at={(1.04,0.0)}},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1157 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1158 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1159 |
xmax=7100000,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1160 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1161 |
ymax=33,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1162 |
%scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1163 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1164 |
width=5cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1165 |
height=5cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1166 |
legend entries={Scala V3, Scala V4},
|
442
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1167 |
legend style={at={(0.1,-0.2)},anchor=north}]
|
436
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1168 |
\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1169 |
\addplot[purple,mark=square*,mark options={fill=white}] table {re4.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1170 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1171 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1172 |
&
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1173 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1174 |
\begin{axis}[
|
442
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1175 |
title={\small{}Graph: \bl{$(a^*)^* \cdot b$}},
|
436
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1176 |
xlabel={$n$},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1177 |
x label style={at={(1.09,0.0)}},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1178 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1179 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1180 |
xmax=8100000,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1181 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1182 |
ymax=33,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1183 |
%scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1184 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1185 |
width=5cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1186 |
height=5cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1187 |
legend entries={Scala V3, Scala V4},
|
442
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1188 |
legend style={at={(0.1,-0.2)},anchor=north}]
|
436
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1189 |
\addplot[black,mark=square*,mark options={fill=white}] table {re3a.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1190 |
\addplot[purple,mark=square*,mark options={fill=white}] table {re4a.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1191 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1192 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1193 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1194 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1195 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1196 |
\only<2->{\footnotesize
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1197 |
\begin{textblock}{9}(0.5,8)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1198 |
\begin{bubble}[11.8cm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1199 |
\lstinputlisting[numbers=none]{../progs/app52.scala}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1200 |
\end{bubble}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1201 |
\end{textblock}}
|
510
|
1202 |
|
436
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1203 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1204 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1205 |
|
638
|
1206 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1207 |
\begin{frame}[c]
|
|
1208 |
\frametitle{Another Homework Question}
|
|
1209 |
|
|
1210 |
\begin{itemize}
|
|
1211 |
\item How many basic regular expressions are there to match
|
|
1212 |
the string \bl{$abcd$}\,?\pause
|
|
1213 |
\item How many if they cannot include
|
|
1214 |
\bl{$\ONE$} and \bl{$\ZERO$}\/?\pause
|
|
1215 |
\item How many if they are also not
|
|
1216 |
allowed to contain stars?\pause
|
|
1217 |
\item How many if they are also
|
|
1218 |
not allowed to contain \bl{$\_ + \_$}\/?
|
|
1219 |
\end{itemize}
|
|
1220 |
|
|
1221 |
\end{frame}
|
|
1222 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
1223 |
|
|
1224 |
|
7
|
1225 |
\end{document}
|
|
1226 |
|
|
1227 |
%%% Local Variables:
|
|
1228 |
%%% mode: latex
|
|
1229 |
%%% TeX-master: t
|
|
1230 |
%%% End:
|
|
1231 |
|