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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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7
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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}{AFL 02, King's College London}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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7
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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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7
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm]
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\LARGE Automata and \\[-2mm]
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Office: & S1.27 (1st floor Strand Building)\\
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Slides: & KEATS
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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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7
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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16
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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{\begin{tabular}{@ {}c@ {}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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An Efficient Regular\\[-1mm]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Expression Matcher\end{tabular}}
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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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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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changeset
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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={\pcode{a}s},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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changeset
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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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changeset
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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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changeset
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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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changeset
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xmax=30,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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changeset
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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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changeset
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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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changeset
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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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changeset
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width=4.5cm,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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changeset
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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]
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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}]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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}]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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changeset
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\begin{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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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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changeset
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xlabel={\pcode{a}s},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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changeset
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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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changeset
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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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changeset
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xtick={0,3000,...,12000},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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xmax=12000,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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ymax=35,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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ytick={0,5,...,30},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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scaled ticks=false,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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axis lines=left,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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width=5.5cm,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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height=4.5cm
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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\addplot[green,mark=square*,mark options={fill=white}] table {re2b.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>
diff
changeset
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\end{axis}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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\end{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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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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changeset
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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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changeset
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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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changeset
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103 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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104 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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105 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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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{Languages}
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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>
diff
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$\{[], hello, \textit{foobar}, a, abc\}$}
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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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16
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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\item \alert{\bf Concatenation} of strings and languages
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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117 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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118 |
\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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changeset
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\begin{tabular}{rcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$\textit{foo}\;@\;bar$} & \bl{$=$} & \bl{$\textit{foobar}$}\medskip\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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>
diff
changeset
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122 |
\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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125 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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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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128 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\[
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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130 |
\bl{A \,@\, B = \{fooa, foob, bara, barb\}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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132 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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133 |
\end{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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134 |
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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>
diff
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137 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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138 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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139 |
\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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\frametitle{The Power Operation}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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141 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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\begin{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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143 |
\item The \alert{\bf Power} of a language:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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144 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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145 |
\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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146 |
\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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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>
diff
changeset
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148 |
\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>
diff
changeset
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149 |
\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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150 |
\end{center}\bigskip
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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151 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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152 |
\item[] For example
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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153 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
154 |
\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
155 |
\begin{tabular}{l}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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156 |
\bl{$A^4 = A \,@\, A \,@\, A \,@\, A$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
157 |
\bl{$A^0 \dn \{[]\}$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
158 |
\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
159 |
\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
160 |
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261
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
161 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
162 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
163 |
\end{frame}
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16
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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165 |
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335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
166 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
167 |
\begin{frame}[c]
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
168 |
\frametitle{Homework Question}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
169 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
170 |
\begin{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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171 |
\item Say \bl{$A = \{[a],[b],[c],[d]\}$}.\bigskip
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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172 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
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173 |
\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
174 |
How many strings are in \bl{$A^4$}?
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
175 |
\end{center}\bigskip\medskip\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
176 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
177 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
178 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
179 |
\begin{tabular}{l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
180 |
What if \bl{$A = \{[a],[b],[c],[]\}$};\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
181 |
how many strings are then in \bl{$A^4$}?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
182 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
183 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
184 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
185 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
186 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
187 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
188 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
189 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
190 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
191 |
\frametitle{The Star Operation}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
192 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
193 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
194 |
\item The \alert{\bf Star} of a \underline{language}:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
195 |
\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
196 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
197 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
198 |
\begin{tabular}{c}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
199 |
\bl{$A^* \dn \bigcup_{0\le n} A^n$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
200 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
201 |
\end{center}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
202 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
203 |
\item[] This expands to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
204 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
205 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
206 |
\bl{A^0 \cup A^1 \cup A^2 \cup A^3 \cup A^4 \cup \ldots}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
207 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
208 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
209 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
210 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
211 |
\bl{\{[]\} \;\cup\; A \;\cup\; A\,@\,A \;\cup\;
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
212 |
A\,@\,A\,@\,A \;\cup\; A\,@\,A\,@\,A\,@\,A \cup \ldots}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
213 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
214 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
215 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
216 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
217 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
218 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
219 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
220 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
221 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
222 |
\frametitle{Semantic Derivative}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
223 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
224 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
225 |
\item The \alert{\bf Semantic Derivative} of a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
226 |
\underline{language}\\ wrt to a character \bl{$c$}:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
227 |
\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
228 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
229 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
230 |
\bl{$Der\,c\,A \dn \{ s \;|\; c\!::\!s \in A\}$ }
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
231 |
\end{center}\bigskip\bigskip\bigskip
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
232 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
233 |
For \bl{$A = \{\textit{foo}, \textit{bar}, \textit{frak}\}$} then
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
234 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
235 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
236 |
\bl{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
237 |
$Der\,f\,A$ & $=$ & $\{\textit{oo}, \textit{rak}\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
238 |
$Der\,b\,A$ & $=$ & $\{\textit{ar}\}$\\
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
239 |
$Der\,a\,A$ & $=$ & $\varnothing$\\\pause
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
240 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
241 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
242 |
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
243 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
244 |
We can extend this definition to strings
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
245 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
246 |
\bl{Ders\,s\,A = \{s'\;|\;s\,@\,s' \in A\}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
247 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
248 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
249 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
250 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
251 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
252 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
253 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
254 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
255 |
\begin{frame}[t]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
256 |
\frametitle{Regular Expressions}
|
|
7
|
257 |
|
|
15
|
258 |
Their inductive definition:
|
|
|
259 |
|
|
|
260 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
261 |
\begin{textblock}{6}(2,7.5)
|
|
7
|
262 |
\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
263 |
\bl{$r$} & \bl{$::=$} & \bl{$\varnothing$} & null\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
264 |
& \bl{$\mid$} & \bl{$\epsilon$} & empty string / \pcode{""} / $[]$\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
265 |
& \bl{$\mid$} & \bl{$c$} & character\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
266 |
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
267 |
& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
268 |
& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
|
|
7
|
269 |
\end{tabular}
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
270 |
\end{textblock}
|
|
7
|
271 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
272 |
\only<2->{\footnotesize
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
273 |
\begin{textblock}{9}(2,0.5)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
274 |
\begin{bubble}[9.8cm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
275 |
\lstinputlisting{../progs/app01.scala}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
276 |
\end{bubble}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
277 |
\end{textblock}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
278 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
279 |
\end{frame}
|
|
16
|
280 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
281 |
|
|
|
282 |
|
|
|
283 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
7
|
284 |
\begin{frame}[c]
|
|
|
285 |
\frametitle{\begin{tabular}{c}The Meaning of a\\[-2mm] Regular Expression\end{tabular}}
|
|
|
286 |
|
|
|
287 |
\begin{textblock}{15}(1,4)
|
|
|
288 |
\begin{tabular}{@ {}rcl}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
289 |
\bl{$L(\varnothing)$} & \bl{$\dn$} & \bl{$\varnothing$}\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
290 |
\bl{$L(\epsilon)$} & \bl{$\dn$} & \bl{$\{[]\}$}\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
291 |
\bl{$L(c)$} & \bl{$\dn$} & \bl{$\{[c]\}$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
292 |
\bl{$L(r_1 + r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \cup L(r_2)$}\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
293 |
\bl{$L(r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
294 |
\bl{$L(r^*)$} & \bl{$\dn$} & \bl{$(L(r))^*$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
295 |
\end{tabular}
|
|
11
|
296 |
\end{textblock}
|
|
7
|
297 |
|
|
16
|
298 |
\begin{textblock}{6}(9,12)\small
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
299 |
\bl{$L$} is a function from regular expressions to
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
300 |
sets of strings\\
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
301 |
\bl{$L$ : Rexp $\Rightarrow$ Set$[$String$]$}
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
302 |
\end{textblock}
|
|
16
|
303 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
304 |
\end{frame}
|
|
7
|
305 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
306 |
|
|
16
|
307 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
308 |
\begin{frame}[c]
|
|
|
309 |
|
|
|
310 |
\large
|
|
|
311 |
\begin{center}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
312 |
What is \bl{$L(a^*)$}?
|
|
16
|
313 |
\end{center}
|
|
|
314 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
315 |
\end{frame}
|
|
16
|
316 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
317 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
318 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
332
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
319 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
320 |
\frametitle{\begin{tabular}{c}
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
321 |
When Are Two Regular\\[-1mm]
|
332
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
322 |
Expressions Equivalent?\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
323 |
\large
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
324 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
325 |
\bl{$r_1 \equiv r_2 \;\;\dn\;\; L(r_1) = L(r_2)$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
326 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
327 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
328 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
329 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
330 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
331 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
332 |
\begin{frame}[t]
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
333 |
\frametitle{Concrete Equivalences}
|
|
16
|
334 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
335 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
336 |
\bl{\begin{tabular}{rcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
337 |
$(a + b) + c$ & $\equiv$ & $a + (b + c)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
338 |
$a + a$ & $\equiv$ & $a$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
339 |
$a + b$ & $\equiv$ & $b + a$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
340 |
$(a \cdot b) \cdot c$ & $\equiv$ & $a \cdot (b \cdot c)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
341 |
$c \cdot (a + b)$ & $\equiv$ & $(c \cdot a) + (c \cdot b)$\bigskip\bigskip\\\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
342 |
$a \cdot a$ & $\not\equiv$ & $a$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
343 |
$a + (b \cdot c)$ & $\not\equiv$ & $(a + b) \cdot (a + c)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
344 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
345 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
346 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
347 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
348 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
349 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
350 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
351 |
\begin{frame}[t]
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
352 |
\frametitle{Corner Cases}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
353 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
354 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
355 |
\bl{\begin{tabular}{rcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
356 |
$a \cdot \varnothing$ & $\not\equiv$ & $a$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
357 |
$a + \epsilon$ & $\not\equiv$ & $a$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
358 |
$\epsilon$ & $\equiv$ & $\varnothing^*$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
359 |
$\epsilon^*$ & $\equiv$ & $\epsilon$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
360 |
$\varnothing^*$ & $\not\equiv$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
361 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
362 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
363 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
364 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
365 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
366 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
367 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
368 |
\begin{frame}[t]
|
336
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
369 |
\frametitle{Simplification Rules}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
370 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
371 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
372 |
\bl{\begin{tabular}{rcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
373 |
$r + \varnothing$ & $\equiv$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
374 |
$\varnothing + r$ & $\equiv$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
375 |
$r \cdot \epsilon$ & $\equiv$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
376 |
$\epsilon \cdot r$ & $\equiv$ & $r$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
377 |
$r \cdot \varnothing$ & $\equiv$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
378 |
$\varnothing \cdot r$ & $\equiv$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
379 |
$r + r$ & $\equiv$ & $r$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
380 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
381 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
382 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
383 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
384 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
16
|
385 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
386 |
%\newcommand{\YES}{\textcolor{gray}{yes}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
387 |
%\newcommand{\NO}{\textcolor{gray}{no}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
388 |
%\newcommand{\FORALLR}{\textcolor{gray}{$\forall$ r.}}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
389 |
|
|
7
|
390 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
391 |
\begin{frame}[c]
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
392 |
\frametitle{\begin{tabular}{c}The Specification\\[-1mm] for Matching\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
393 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
394 |
\large
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
395 |
A regular expression \bl{$r$} matches a string \bl{$s$}\\ if and only if
|
|
7
|
396 |
|
|
|
397 |
\begin{center}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
398 |
\bl{$s \in L(r)$}\\
|
|
7
|
399 |
\end{center}
|
|
|
400 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
401 |
\end{frame}
|
|
7
|
402 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
403 |
|
|
|
404 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
405 |
\begin{frame}[c]
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
406 |
\frametitle{\bl{$(a^?^{\{n\}}) \cdot a^{\{n\}}$}}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
407 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
408 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
409 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
410 |
\begin{axis}[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
411 |
xlabel={\pcode{a}s},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
412 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
413 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
414 |
xtick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
415 |
xmax=30,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
416 |
ymax=35,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
417 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
418 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
419 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
420 |
width=9cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
421 |
height=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
422 |
legend entries={Python,Ruby},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
423 |
legend pos=north west,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
424 |
legend cell align=left
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
425 |
]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
426 |
\addplot[blue,mark=*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
427 |
table {re-python.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
428 |
\addplot[brown,mark=pentagon*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
429 |
table {re-ruby.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
430 |
\end{axis}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
431 |
\end{tikzpicture}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
432 |
\end{center}
|
|
11
|
433 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
434 |
\end{frame}
|
|
11
|
435 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
436 |
|
|
|
437 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
118
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
438 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
439 |
\frametitle{Evil Regular Expressions}
|
118
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
440 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
441 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
442 |
\item \alert{R}egular \alert{e}xpression \alert{D}enial \alert{o}f \alert{S}ervice (ReDoS)\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
443 |
\item Evil regular expressions\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
444 |
\begin{itemize}
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
445 |
\item \bl{$(a^?^{\{n\}}) \cdot a^{\{n\}}$}
|
118
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
446 |
\item \bl{$(a^+)^+$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
447 |
\item \bl{$([a$\,-\,$z]^+)^*$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
448 |
\item \bl{$(a + a \cdot a)^+$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
449 |
\item \bl{$(a + a?)^+$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
450 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
451 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
452 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
453 |
\end{frame}
|
118
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
454 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
455 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
456 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
11
|
457 |
\begin{frame}[t]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
458 |
\frametitle{A Matching Algorithm}
|
|
11
|
459 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
460 |
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
461 |
\ldots{}whether a regular expression can match the empty string:
|
|
11
|
462 |
\begin{center}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
463 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
464 |
\bl{$nullable(\varnothing)$} & \bl{$\dn$} & \bl{\textit{false}}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
465 |
\bl{$nullable(\epsilon)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
466 |
\bl{$nullable (c)$} & \bl{$\dn$} & \bl{\textit{false}}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
467 |
\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
|
468 |
\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
|
469 |
\bl{$nullable (r^*)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
470 |
\end{tabular}
|
|
16
|
471 |
\end{center}
|
|
|
472 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
473 |
\end{frame}
|
|
11
|
474 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
475 |
|
|
|
476 |
|
|
|
477 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
478 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
479 |
\frametitle{The Derivative of a Rexp}
|
|
7
|
480 |
|
|
|
481 |
\large
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
482 |
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
|
483 |
expression that matches just \bl{$s$}?\bigskip\bigskip\bigskip\bigskip
|
|
7
|
484 |
|
|
|
485 |
\small
|
325
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
486 |
\bl{$der\,c\,r$} gives the answer, Brzozowski 1964
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
487 |
\end{frame}
|
|
7
|
488 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
489 |
|
|
|
490 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
491 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
492 |
\frametitle{The Derivative of a Rexp}
|
|
7
|
493 |
|
|
|
494 |
\begin{center}
|
|
|
495 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
117
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
496 |
\bl{$der\, c\, (\varnothing)$} & \bl{$\dn$} & \bl{$\varnothing$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
497 |
\bl{$der\, c\, (\epsilon)$} & \bl{$\dn$} & \bl{$\varnothing$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
498 |
\bl{$der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\epsilon$ else $\varnothing$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
499 |
\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
|
500 |
\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
|
501 |
& & \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
|
502 |
& & \bl{else $(der\, c\, r_1) \cdot r_2$}\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
503 |
\bl{$der\, c\, (r^*)$} & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\bigskip\\\pause
|
|
7
|
504 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
505 |
\bl{$\textit{ders}\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
506 |
\bl{$\textit{ders}\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\textit{ders}\,s\,(der\,c\,r)$} & \\
|
|
7
|
507 |
\end{tabular}
|
|
|
508 |
\end{center}
|
|
|
509 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
510 |
\end{frame}
|
|
7
|
511 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
512 |
|
|
|
513 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
514 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
515 |
\frametitle{Examples}
|
|
7
|
516 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
517 |
Given \bl{$r \dn ((a \cdot b) + b)^*$} what is
|
|
7
|
518 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
519 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
520 |
\begin{tabular}{l}
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
521 |
\bl{$der\,a\,r =\;?$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
522 |
\bl{$der\,b\,r =\;?$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
523 |
\bl{$der\,c\,r =\;?$}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
524 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
525 |
\end{center}
|
|
7
|
526 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
527 |
\end{frame}
|
|
7
|
528 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
529 |
|
|
17
|
530 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
531 |
\begin{frame}[t]
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
532 |
\frametitle{The Algorithm}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
533 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
534 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
535 |
\begin{tabular}{@{}rll@{}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
536 |
Input: & \bl{$r_1$}, \bl{$abc$}\medskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
537 |
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
|
538 |
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
|
539 |
Step 3: & build derivative of \bl{$c$} and \bl{$r_3$} & \bl{$(r_4 = der\,b\,r_3)$}\smallskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
540 |
Step 4: & the string is exhausted; test & (\bl{$nullable(r_4)$})\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
541 |
& whether \bl{$r_4$} can recognise\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
542 |
& the empty string\smallskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
543 |
Output: & result of the test\\
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
544 |
& $\Rightarrow \bl{\textit{true}} \,\text{or}\,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
545 |
\bl{\textit{false}}$\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
546 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
547 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
548 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
549 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
550 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
551 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
552 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
553 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
554 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
555 |
\frametitle{The Idea of the Algorithm}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
556 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
557 |
If we want to recognise the string \bl{$abc$} with regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
558 |
expression \bl{$r_1$} then\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
559 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
560 |
\begin{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
561 |
\item \bl{$Der\,a\,(L(r_1))$}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
562 |
\item \bl{$Der\,b\,(Der\,a\,(L(r_1)))$}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
563 |
\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
|
564 |
\item finally we test whether the empty string is in this
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
565 |
set; same for \bl{$Ders\,abc\,(L(r_1))$}.\medskip
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
566 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
567 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
568 |
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
|
569 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
570 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
571 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
572 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
573 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
574 |
\begin{frame}[c]
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
575 |
\frametitle{\bl{$(a^?^{\{n\}}) \cdot a^{\{n\}}$}}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
576 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
577 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
578 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
579 |
\begin{axis}[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
580 |
xlabel={\pcode{a}s},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
581 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
582 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
583 |
xtick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
584 |
xmax=30,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
585 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
586 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
587 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
588 |
width=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
589 |
height=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
590 |
legend entries={Python,Ruby,Scala V1},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
591 |
legend pos=outer north east,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
592 |
legend cell align=left
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
593 |
]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
594 |
\addplot[blue,mark=*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
595 |
table {re-python.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
596 |
\addplot[brown,mark=pentagon*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
597 |
table {re-ruby.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
598 |
\addplot[red,mark=triangle*,mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
599 |
table {re1.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
600 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
601 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
602 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
603 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
604 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
605 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
606 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
607 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
608 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
609 |
\frametitle{A Problem}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
610 |
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
611 |
We represented the ``n-times'' \bl{$a^{\{n\}}$} as a
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
612 |
sequence regular expression:
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
613 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
614 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
615 |
\begin{tabular}{rl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
616 |
1: & \bl{$a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
617 |
2: & \bl{$a\cdot a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
618 |
3: & \bl{$a\cdot a\cdot a$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
619 |
& \ldots\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
620 |
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
|
621 |
& \ldots\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
622 |
20:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
623 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
624 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
625 |
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
626 |
This problem is aggravated with \bl{$a^?$} being represented
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
627 |
as \bl{$\epsilon + a$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
628 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
629 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
630 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
631 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
632 |
\begin{frame}[c]
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
633 |
\frametitle{Solving the Problem}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
634 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
635 |
What happens if we extend our regular expressions
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
636 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
637 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
638 |
\begin{tabular}{rcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
639 |
\bl{$r$} & \bl{$::=$} & \bl{\ldots}\\
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
640 |
& \bl{$\mid$} & \bl{$r^{\{n\}}$}\\
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
641 |
& \bl{$\mid$} & \bl{$r^?$}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
642 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
643 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
644 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
645 |
What is their meaning? What are the cases for \bl{$nullable$} and \bl{$der$}?
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
646 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
647 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
648 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
649 |
|
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 |
\begin{frame}[t]
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
652 |
\frametitle{\bl{$(a^?^{\{n\}}) \cdot a^{\{n\}}$}}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
653 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
654 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
655 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
656 |
\begin{axis}[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
657 |
xlabel={\pcode{a}s},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
658 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
659 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
660 |
xtick={0,200,...,1000},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
661 |
xmax=1000,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
662 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
663 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
664 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
665 |
width=9.5cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
666 |
height=7cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
667 |
legend entries={Python,Ruby,Scala V1,Scala V2},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
668 |
legend pos=north west,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
669 |
legend cell align=left
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
670 |
]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
671 |
\addplot[blue,mark=*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
672 |
table {re-python.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
673 |
\addplot[brown,mark=pentagon*, mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
674 |
table {re-ruby.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
675 |
\addplot[red,mark=triangle*,mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
676 |
table {re1.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
677 |
\addplot[green,mark=square*,mark options={fill=white}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
678 |
table {re2b.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
679 |
\end{axis}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
680 |
\end{tikzpicture}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
681 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
682 |
|
335
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
683 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
684 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
685 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
686 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
687 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
688 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
689 |
\frametitle{Examples}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
690 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
691 |
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
|
692 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
693 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
694 |
\begin{tabular}{l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
695 |
\bl{$der\,a\,r = ((\epsilon \cdot b) + \varnothing) \cdot r$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
696 |
\bl{$der\,b\,r = ((\varnothing \cdot b) + \epsilon)\cdot r$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
697 |
\bl{$der\,c\,r = ((\varnothing \cdot b) + \varnothing)\cdot r$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
698 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
699 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
700 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
701 |
What are these regular expressions equivalent to?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
702 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
703 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
704 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
705 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
706 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
707 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
708 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
709 |
\begin{frame}[t]
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
710 |
\frametitle{\bl{$(a^?^{\{n\}}) \cdot a^{\{n\}}$}}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
711 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
712 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
713 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
714 |
\begin{axis}[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
715 |
xlabel={\pcode{a}s},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
716 |
ylabel={time in secs},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
717 |
enlargelimits=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
718 |
xtick={0,3000,...,12000},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
719 |
xmax=12000,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
720 |
ymax=35,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
721 |
ytick={0,5,...,30},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
722 |
scaled ticks=false,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
723 |
axis lines=left,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
724 |
width=9cm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
725 |
height=7cm
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
726 |
]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
727 |
\addplot[green,mark=square*,mark options={fill=white}] table {re2b.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
728 |
\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
729 |
\end{axis}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
730 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
731 |
\end{center}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
732 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
733 |
\end{frame}
|
119
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 |
|
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{frame}[t]
|
342
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
738 |
\frametitle{What is good about this Alg.}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
739 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
740 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
741 |
\item extends to most regular expressions, for example
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
742 |
\bl{$\sim r$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
743 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
744 |
\item is easy to implement in a functional language
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
745 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
746 |
\item the algorithm is already quite old; there is still
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
747 |
work to be done to use it as a tokenizer (that is brand new work)
|
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 |
\item we can prove its correctness\ldots
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
750 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
751 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
752 |
\end{frame}
|
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 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
755 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
756 |
\begin{frame}[t]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
757 |
\frametitle{Proofs about Rexps}
|
|
17
|
758 |
|
|
|
759 |
Remember their inductive definition:\\[5cm]
|
|
|
760 |
|
|
|
761 |
\begin{textblock}{6}(5,5)
|
|
|
762 |
\begin{tabular}{@ {}rrl}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
763 |
\bl{$r$} & \bl{$::=$} & \bl{$\varnothing$}\\
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
764 |
& \bl{$\mid$} & \bl{$\epsilon$} \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
765 |
& \bl{$\mid$} & \bl{$c$} \\
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
766 |
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$}\\
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
767 |
& \bl{$\mid$} & \bl{$r_1 + r_2$} \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
768 |
& \bl{$\mid$} & \bl{$r^*$} \\
|
|
17
|
769 |
\end{tabular}
|
|
|
770 |
\end{textblock}
|
|
|
771 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
772 |
If we want to prove something, say a property \bl{$P(r)$}, for all regular expressions \bl{$r$} then \ldots
|
|
17
|
773 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
774 |
\end{frame}
|
|
17
|
775 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
776 |
|
|
|
777 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
778 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
779 |
\frametitle{Proofs about Rexp (2)}
|
|
17
|
780 |
|
|
|
781 |
\begin{itemize}
|
|
|
782 |
\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
783 |
\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
|
784 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
785 |
\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
|
786 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
787 |
\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
|
788 |
holds for \bl{$r$}.
|
|
17
|
789 |
\end{itemize}
|
|
|
790 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
791 |
\end{frame}
|
|
17
|
792 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
793 |
|
|
|
794 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
795 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
796 |
\frametitle{Proofs about Rexp (3)}
|
|
17
|
797 |
|
|
|
798 |
Assume \bl{$P(r)$} is the property:
|
|
|
799 |
|
|
|
800 |
\begin{center}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
801 |
\bl{$nullable(r)$} if and only if \bl{$[] \in L(r)$}
|
|
17
|
802 |
\end{center}
|
|
|
803 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
804 |
\end{frame}
|
|
17
|
805 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
806 |
|
|
|
807 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
808 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
809 |
\frametitle{Proofs about Rexp (4)}
|
263
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 |
\begin{center}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
812 |
\bl{\begin{tabular}{r@{\hspace{2mm}}c@{\hspace{2mm}}l}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
813 |
$rev(\varnothing)$ & $\dn$ & $\varnothing$\\
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
814 |
$rev(\epsilon)$ & $\dn$ & $\epsilon$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
815 |
$rev(c)$ & $\dn$ & $c$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
816 |
$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
817 |
$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
818 |
$rev(r^*)$ & $\dn$ & $rev(r)^*$\\
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
819 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
820 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
821 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
822 |
We can prove
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
823 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
824 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
825 |
\bl{$L(rev(r)) = \{s^{-1} \;|\; s \in L(r)\}$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
826 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
827 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
828 |
by induction on \bl{$r$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
829 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
830 |
\end{frame}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
831 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
832 |
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
833 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
834 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
835 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
836 |
\frametitle{\begin{tabular}{c}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
837 |
Correctness Proof\\[-1mm] for our Matcher\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
838 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
839 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
840 |
\item We started from
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
841 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
842 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
843 |
\begin{tabular}{rp{4cm}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
844 |
& \bl{$s \in L(r)$}\medskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
845 |
\bl{$\Leftrightarrow$} & \bl{$[] \in Ders\,s\,(L(r))$}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
846 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
847 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
848 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
849 |
\item if we can show \bl{$Ders\, s\,(L(r)) = L(ders\,s\,r)$} we
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
850 |
have
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
851 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
852 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
853 |
\begin{tabular}{rp{4cm}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
854 |
\bl{$\Leftrightarrow$} & \bl{$[] \in L(ders\,s\,r)$}\medskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
855 |
\bl{$\Leftrightarrow$} & \bl{$nullable(ders\,s\,r)$}\medskip\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
856 |
\bl{$\dn$} & \bl{$matches\,s\,r$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
857 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
858 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
859 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
860 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
861 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
862 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
863 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
864 |
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
865 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
866 |
\begin{frame}[c]
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
867 |
\frametitle{Proofs about Rexp (5)}
|
263
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
868 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
869 |
Let \bl{$Der\,c\,A$} be the set defined as
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
870 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
871 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
872 |
\bl{$Der\,c\,A \dn \{ s \;|\; c\!::\!s \in A\}$ }
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
873 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
874 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
875 |
We can prove
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
876 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
877 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
878 |
\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
879 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
880 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
881 |
by induction on \bl{$r$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
882 |
|
337
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
883 |
\end{frame}
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
884 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
885 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
886 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
887 |
\begin{frame}[c]
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
888 |
\frametitle{Proofs about Strings}
|
|
17
|
889 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
890 |
If we want to prove something, say a property \bl{$P(s)$}, for all strings \bl{$s$} then \ldots\bigskip
|
|
17
|
891 |
|
|
|
892 |
\begin{itemize}
|
|
|
893 |
\item \bl{$P$} holds for the empty string, and\medskip
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
894 |
\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
|
895 |
already holds for \bl{$s$}
|
|
17
|
896 |
\end{itemize}
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
897 |
\end{frame}
|
|
17
|
898 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
899 |
|
|
19
|
900 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
901 |
\begin{frame}[c]
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
902 |
\frametitle{Proofs about Strings (2)}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
903 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
904 |
We can then prove
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
905 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
906 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
907 |
\bl{$Ders\,s\,(L(r)) = L(ders\,s\,r)$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
908 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
909 |
|
|
19
|
910 |
|
119
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
911 |
We can finally prove
|
|
19
|
912 |
|
|
|
913 |
\begin{center}
|
345
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
914 |
\bl{$matches\,s\,r$} if and only if \bl{$s \in L(r)$}
|
|
19
|
915 |
\end{center}
|
|
|
916 |
|
|
|
917 |
|
338
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
918 |
\end{frame}
|
|
19
|
919 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
17
|
920 |
|
|
7
|
921 |
\end{document}
|
|
|
922 |
|
|
|
923 |
%%% Local Variables:
|
|
|
924 |
%%% mode: latex
|
|
|
925 |
%%% TeX-master: t
|
|
|
926 |
%%% End:
|
|
|
927 |
|