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% !TEX program = xelatex
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\documentclass[dvipsnames,14pt,t,xelatex,xcolor={table}]{beamer}
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\usepackage{../slides}
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\usepackage{../graphicss}
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\usepackage{../langs}
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\usepackage{../data}
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\usetikzlibrary{cd}
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\usepackage{tcolorbox}
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\newtcolorbox{mybox}{colback=red!5!white,colframe=red!75!black}
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\newtcolorbox{mybox2}[1]{colback=red!5!white,colframe=red!75!black,fonttitle=\bfseries,title=#1}
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\newtcolorbox{mybox3}[1]{colback=Cyan!5!white,colframe=Cyan!75!black,fonttitle=\bfseries,title=#1}
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\hfuzz=220pt
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\lstset{language=Scala,
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style=mystyle,
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numbersep=0pt,
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numbers=none,
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xleftmargin=0mm}
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\pgfplotsset{compat=1.17}
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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% beamer stuff
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\renewcommand{\slidecaption}{SSY, King's College London}
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\begin{document}
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876
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[8mm]
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\LARGE Derivative-Based\\
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\LARGE Regular Expression Matching\\[5mm]
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\normalsize\textcolor{gray}{Christian Urban, SSY Group}
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\end{tabular}}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{A Bit of Background}
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\begin{textblock}{10}(2,2.5)
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\includegraphics[scale=0.4]{../pics/phd-1.jpg}
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\end{textblock}
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\begin{textblock}{11}(1.7,10)
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\begin{bubble}[9.6cm]
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PhD thesis on a particular term-rewriting system:\medskip\\
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Proved that all rewriting sequences are strongly normalising (terminate).
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\end{bubble}
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\end{textblock}
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\only<2->{%
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\begin{textblock}{3}(10,2.5)
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\begin{bubble}[3cm]
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\begin{tabular}{rcl}
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\bl{$t$} & \bl{::=} & \bl{$x$}\\
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& \bl{|} & \bl{$\lambda x.\,t$}\\
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& \bl{|} & \bl{$t_1\;t_2$}\\
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& \bl{|} & \bl{...}
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\end{tabular}
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\end{bubble}
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\end{textblock}}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Quiz 1}
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\begin{bubble}[10cm]\it
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There are many, many regular expression libraries.\bigskip\\
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\textbf{Given a regular expression \bl{r} and a string \bl{s}, what is the
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difficulty / complexity of the problem deciding whether \bl{r} matches \bl{s}?}
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\end{bubble}
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\begin{textblock}{12}(2,9)
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\begin{tikzpicture}
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\draw(0,0) node[rotate=20]{linear};
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\draw(3,0) node[rotate=-20]{\bl{$O(n \log n)$}};
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\draw(3,-2) node[rotate=10]{quadratic};
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\draw(6,0) node[rotate=20]{P};
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\draw(7,0) node[rotate=-20]{NP};
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\draw(7,-1) node[rotate=0]{\bl{$O(2^n)$}};
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\end{tikzpicture}
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Quiz 2}
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\begin{textblock}{12}(2,2.5)
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A regular expression for (some) email addresses:
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\begin{center}
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($\underbrace{\bl{[a\mbox{-}z0\mbox{-}9\_\!\!\_\,.-]^+}}_{\textrm{name}}$)\bl{$\,@\,$}
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($\underbrace{\bl{[a\mbox{-}z0\mbox{-}9\,-]^+}}_{\textrm{domain}}$) \bl{$\,.\,$}
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($\underbrace{\bl{[a\mbox{-}z\,.]^{\{2..5\}}}}_{\textrm{top-level domain}}$)
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\end{center}
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\end{textblock}
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\only<2>{
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\begin{textblock}{10}(4,8)
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bounded regular expressions:\medskip\\
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\begin{tabular}{ll}
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\bl{$r^{\{n\}}$} & exactly n-times \bl{$r$}\\
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\bl{$r^{\{n..m\}}$} & between n and m-times\\
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\bl{$r^{\{n..\}}$} & from n-times\\
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\bl{$r^{\{..m\}}$} & upto m-times\\
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\end{tabular}\smallskip\\
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\footnotesize\hfill ${}^*$ in some applications the counters can be in the millions
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\end{textblock}}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Quiz 2}
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\begin{textblock}{12}(2,2.5)
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\onslide<1->{Thompson construction: By recursion we are given two NFAs for $r_1$ and $r_2$.\medskip\\
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\onslide<2->{For $r_1\cdot r_2$:}\bigskip}
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\end{textblock}
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\begin{textblock}{12}(2,6)
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\onslide<1>{
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\begin{tikzpicture}[node distance=3mm,
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>=stealth',very thick,
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every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20}]
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\node[state, initial] (Q_0) {$\mbox{}$};
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\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
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\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
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\node (R_1) [right=of Q_0] {$\ldots$};
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\node[state, accepting] (T_1) [right=of R_1] {$\mbox{}$};
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\node[state, accepting] (T_2) [above=of T_1] {$\mbox{}$};
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\node[state, accepting] (T_3) [below=of T_1] {$\mbox{}$};
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\node (A_0) [right=2.5cm of T_1] {$\mbox{}$};
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\node[state, initial] (A_01) [above=1mm of A_0] {$\mbox{}$};
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\node[state, initial] (A_02) [below=1mm of A_0] {$\mbox{}$};
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\node (b_1) [right=of A_0] {$\ldots$};
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\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
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\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
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\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
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\begin{pgfonlayer}{background}
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\node (1) [rounded corners, inner sep=1mm, thick,
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draw=black!60, fill=black!20, fit= (Q_0) (R_1) (T_1) (T_2) (T_3)] {};
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\node (2) [rounded corners, inner sep=1mm, thick,
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draw=black!60, fill=black!20, fit= (A_0) (b_1) (c_1) (c_2) (c_3)] {};
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\node [yshift=2mm] at (1.north) {$r_1$};
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\node [yshift=2mm] at (2.north) {$r_2$};
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\end{pgfonlayer}
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\end{tikzpicture}}
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\end{textblock}
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\begin{textblock}{12}(2,6)
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\onslide<2>{
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\begin{tikzpicture}[node distance=3mm,
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>=stealth',very thick,
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every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
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\node[state, initial] (Q_0) {$\mbox{}$};
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\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
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\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
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\node (r_1) [right=of Q_0] {$\ldots$};
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\node[state] (t_1) [right=of r_1] {$\mbox{}$};
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\node[state] (t_2) [above=of t_1] {$\mbox{}$};
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\node[state] (t_3) [below=of t_1] {$\mbox{}$};
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\node (A_0) [right=2.5cm of t_1] {$\mbox{}$};
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\node[state] (A_01) [above=1mm of A_0] {$\mbox{}$};
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\node[state] (A_02) [below=1mm of A_0] {$\mbox{}$};
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\node (b_1) [right=of A_0] {$\ldots$};
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\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
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\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
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\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
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\path[->] (t_1) edge (A_01);
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\path[->] (t_2) edge node [above] {$\varepsilon$\footnotesize{}s} (A_01);
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\path[->] (t_3) edge (A_01);
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\path[->] (t_1) edge (A_02);
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\path[->] (t_2) edge (A_02);
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\path[->] (t_3) edge node [below] {$\varepsilon$\footnotesize{}s} (A_02);
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\begin{pgfonlayer}{background}
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\node (3) [rounded corners, inner sep=1mm, thick,
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draw=black!60, fill=black!20, fit= (Q_0) (c_1) (c_2) (c_3)] {};
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\node [yshift=2mm] at (3.north) {$r_1\cdot r_2$};
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\end{pgfonlayer}
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\end{tikzpicture}}
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Quiz 2}
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\begin{textblock}{12}(2,2.5)
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\onslide<1->{Thompson construction: By recursion we have a NFA for $r$.\medskip\\
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\onslide<2->{For $r^*$:\bigskip}}
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\end{textblock}
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\begin{textblock}{12}(4,6)
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\onslide<1>{
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\begin{tikzpicture}[node distance=3mm,
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>=stealth',very thick,
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every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
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baseline=(current bounding box.north)]
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\node (2) {$\mbox{}$};
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\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
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\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
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\node (a) [right=of 2] {$\ldots$};
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\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
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\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
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\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
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\begin{pgfonlayer}{background}
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\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
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\node [yshift=3mm] at (1.north) {$r$};
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\end{pgfonlayer}
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\end{tikzpicture}}
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\end{textblock}
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\begin{textblock}{12}(2,6)
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\onslide<2->{
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\begin{tikzpicture}[node distance=3mm,
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>=stealth',very thick,
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every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
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baseline=(current bounding box.north)]
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\node at (0,0) [state, initial,accepting] (1) {$\mbox{}$};
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\node (2) [right=16mm of 1] {$\mbox{}$};
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\node[state] (4) [above=1mm of 2] {$\mbox{}$};
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\node[state] (5) [below=1mm of 2] {$\mbox{}$};
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\node (a) [right=of 2] {$\ldots$};
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\node[state] (a1) [right=of a] {$\mbox{}$};
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\node[state] (a2) [above=of a1] {$\mbox{}$};
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\node[state] (a3) [below=of a1] {$\mbox{}$};
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\path[->] (1) edge node [below] {$\varepsilon$} (4);
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\path[->] (1) edge node [below] {$\varepsilon$} (5);
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\path[->] (a1) edge [bend left=45] node [below] {$\varepsilon$} (1);
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\path[->] (a2) edge [bend right] node [below] {$\varepsilon$} (1);
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\path[->] (a3) edge [bend left=45] node [below] {$\varepsilon$} (1);
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\begin{pgfonlayer}{background}
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\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3)] {};
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\node [yshift=3mm] at (2.north) {$r^*$};
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\end{pgfonlayer}
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\end{tikzpicture}}
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\end{textblock}
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\begin{textblock}{12}(2,12)
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\onslide<3->{
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\begin{bubble}[9.5cm]\it
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Quiz:\smallskip\\
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\textbf{What is the Thompson Construction for \bl{$r^{\{n\}}$} ?}
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\end{bubble}}
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Quiz 3}
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Hierarchy of Languages:
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\begin{center}
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\begin{tikzpicture}
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[rect/.style={draw=black!50,
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top color=white,
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bottom color=black!20,
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rectangle,
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very thick,
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rounded corners}, scale=1.2]
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\draw (0,0) node [rect, text depth=39mm, text width=68mm] {all languages};
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\draw (0,-0.4) node [rect, text depth=28.5mm, text width=64mm] {decidable languages};
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\draw (0,-0.85) node [rect, text depth=17mm] {\;\;context sensitive languages\;\;};
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\draw (0,-1.14) node [rect, text depth=9mm, text width=50mm] {context-free languages};
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\draw (0,-1.4) node [rect] {regular languages};
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\end{tikzpicture}
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\end{center}
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\begin{textblock}{12}(2,11.5)
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\onslide<2->{
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\begin{bubble}[9.5cm]\it
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Quiz:\smallskip\\
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\textbf{Can we use standard parsing algorithms for matching / lexing\,?}
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Say CYK, LL, LR(k), PEG, \ldots
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\end{bubble}}
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\defverbatim{\foo}{\footnotesize%
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\begin{verbatim}
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(?:(?:\"|'|\]|\}|\\|\d|(?:nan|infinity|true|false|
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null|undefined|symbol|math)|\`|\-|\+)+[)]*;?((?:\s
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| -|~|!|{}|\|\||\+)*.*(?:.*=.*)))
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\end{verbatim}
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}
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\begin{frame}[t,fragile]
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\frametitle{Quiz 4}
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\begin{textblock}{12}(2,2.5)
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\begin{bubble}[9.5cm]\it
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Quiz:\smallskip\\
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Do regular expressions have any security relevance\,?
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\end{bubble}
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\end{textblock}
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\only<2->{
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\begin{textblock}{2}(1,6)
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\includegraphics[scale=0.8]{../pics/zeek.png}
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\includegraphics[scale=0.17]{../pics/snort.png}
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\end{textblock}}
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\only<3->{
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\begin{textblock}{7}(7,5.6)
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\includegraphics[scale=0.14]{../pics/cloudflare.png}\\
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\footnotesize
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It serves more web traffic than Twitter, Amazon, Apple,
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Instagram, Bing \& Wikipedia combined.\medskip\\
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Web Application Firewall filters out
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SQL injection attacks, XSS attacks etc
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\end{textblock}}
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\only<3->{
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\begin{textblock}{13}(4,12.3)
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\footnotesize
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\hspace{1.5cm}a global outage on 2 July 2019 (first one for six years)
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\color{blue}\foo\color{black}
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\end{textblock}
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Quiz 3}
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\begin{textblock}{12}(2,2)
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\begin{bubble}[9.5cm]\it
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Quiz:\smallskip\\
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\textbf{Can we use standard parsing algorithms for matching / lexing\,?}
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Say CYK, LL, LR(k), PEG, \ldots
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\end{bubble}
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\end{textblock}
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\begin{textblock}{12}(1,7)
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\textbf{POSIX lexing}
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\begin{tabular}{@ {}ll}
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Assume you have: & \bl{$r_{key} \dn \texttt{if} + \texttt{then} + \texttt{while} \ldots$}\\
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& \bl{$r_{id\;} \,\dn \texttt{[a-z]} \cdot \texttt{[a-z0-9\_]}^*$}\medskip\\
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Lexing a string with & \bl{$(r_{key} + r_{id})^*$}\medskip\\
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What should be & \\
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the result for: & \bl{\texttt{"iffoo"}} \;\;\; \bl{\texttt{"if"}}\\
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\end{tabular}
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\end{textblock}
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\only<2->{
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\begin{textblock}{13.5}(1.5,4)
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\begin{mybox3}{POSIX rules}
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\begin{itemize}
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\item \textbf{The Longest Match Rule:} The longest initial
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substring matched by any regular expression is taken as next token.
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\item \textbf{Priority Rule:} For a particular longest initial
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substring, the first (leftmost) regular expression that can match
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determines the token.
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\item \ldots
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\end{itemize}
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\begin{center}
|
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\bl{$(a + ab) \cdot (bc + c)$} \;\;and\;\; \bl{\texttt{"}$abc$\texttt{"}}
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\end{center}
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|
410 |
\end{mybox3}
|
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876
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\end{textblock}}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
913
|
416 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
417 |
\begin{frame}[t]
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|
418 |
\frametitle{Quiz 1}
|
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\begin{bubble}[10cm]\it
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420 |
There are many, many regular expression libraries.\bigskip\\
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|
421 |
|
|
|
422 |
\textbf{Given a regular expression \bl{r} and a string \bl{s}, what is the
|
|
|
423 |
difficulty / complexity of the problem deciding whether \bl{r} matches \bl{s}?}
|
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|
424 |
\end{bubble}
|
|
|
425 |
|
|
|
426 |
\begin{textblock}{12}(1,9)
|
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427 |
\begin{tabular}{p{4cm}|p{4cm}|p{2cm}}
|
|
|
428 |
atrociously slow (s't) & pretty lazy (s't) & \textcolor{gray}{OTBT}\medskip\\
|
|
|
429 |
Python, Ruby, Swift, Dart, JavaScript & Rust, Go, RE2 & \textcolor{gray}{Snort, .Net7}\\
|
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|
430 |
\end{tabular}
|
|
|
431 |
\end{textblock}
|
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432 |
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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435 |
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436 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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438 |
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439 |
\mbox{}\\[10mm]
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440 |
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\begin{columns}[t,onlytextwidth]
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442 |
\begin{column}{.2\textwidth}
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443 |
\raisebox{-10mm}{
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|
|
444 |
\begin{tikzpicture}
|
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445 |
\begin{axis}[
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|
446 |
xlabel={\bl{$n$} \pcode{a}s},
|
|
|
447 |
%%x label style={at={(1.05,0.0)}},
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448 |
ylabel={time in secs},
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449 |
enlargelimits=false,
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xtick={0,10,...,30},
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451 |
xmax=35,
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ymax=35,
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ytick={0,5,...,30},
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scaled ticks=false,
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455 |
axis lines=left,
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456 |
width=5cm,
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|
457 |
height=5cm,
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|
458 |
legend entries={Python,JavaScript,Swift,Dart},
|
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|
459 |
legend style={font=\small,at={(0.5,-0.39)},anchor=north},
|
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460 |
legend cell align=left]
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|
461 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
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462 |
\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
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|
463 |
\addplot[magenta,mark=*, mark options={fill=white}] table {re-swift.data};
|
|
|
464 |
\addplot[brown,mark=*, mark options={fill=white}] table {re-dart.data};
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|
465 |
\end{axis}
|
|
|
466 |
\end{tikzpicture}}
|
|
|
467 |
\end{column}
|
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|
468 |
|
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|
469 |
\begin{column}{.2\textwidth}
|
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|
470 |
\begin{tikzpicture}
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|
471 |
\begin{axis}[
|
|
|
472 |
xlabel={\bl{$n$} \pcode{a}s},
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%ylabel={time in secs},
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474 |
enlargelimits=false,
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|
475 |
xtick={0,10,...,30},
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476 |
xmax=35,
|
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|
477 |
ymax=35,
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|
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ytick={0,5,...,30},
|
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|
479 |
scaled ticks=false,
|
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|
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axis lines=left,
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481 |
width=5cm,
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482 |
height=5cm,
|
|
|
483 |
legend entries={Python,Ruby},
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|
484 |
legend style={font=\small,at={(0.5,-0.39)},anchor=north},
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legend cell align=left]
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|
486 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
|
|
|
487 |
\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};
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|
|
488 |
\end{axis}
|
|
|
489 |
\end{tikzpicture}
|
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|
490 |
\end{column}
|
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|
491 |
\end{columns}
|
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|
492 |
|
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|
493 |
\begin{textblock}{4}(9,1.7)
|
|
|
494 |
\textbf{\bl{\texttt{(a?)\boldmath$^{\{n\}}\cdot$(a)\boldmath$^{\{n\}}$}}}
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|
495 |
\end{textblock}
|
|
|
496 |
|
|
|
497 |
\begin{textblock}{4}(4,1.7)
|
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|
498 |
\textbf{\bl{\texttt{(a*)*$\cdot$b}}}
|
|
|
499 |
\end{textblock}
|
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|
500 |
|
|
|
501 |
\begin{textblock}{3.4}(0.3,10)
|
|
|
502 |
\small{}\textbf{matching with strings}
|
|
|
503 |
\textbf{\bl{$\underbrace{\texttt{a}...\texttt{a}}_n$}}
|
|
|
504 |
\end{textblock}
|
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505 |
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|
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\end{frame}
|
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507 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
508 |
|
|
|
509 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
510 |
\begin{frame}[t]
|
|
|
511 |
\frametitle{Quiz 1}
|
|
|
512 |
\begin{bubble}[10cm]\it
|
|
|
513 |
\textbf{Given a regular expression \bl{r} and a string \bl{s}, what is the
|
|
|
514 |
difficulty / complexity of the problem deciding whether \bl{r} matches \bl{s}?}
|
|
|
515 |
\end{bubble}
|
|
|
516 |
|
|
|
517 |
\begin{textblock}{12}(1.7,7)
|
|
|
518 |
For \textbf{Perl-style} regular expression matchers (say Python, JavaScript, etc), the
|
|
|
519 |
answer is:
|
|
|
520 |
\only<2->{\begin{center}
|
|
|
521 |
\fontspec{Hoefler Text Black}\textcolor{ProcessBlue}{\huge{}NP}
|
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|
522 |
\end{center}}
|
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|
523 |
\end{textblock}
|
|
|
524 |
|
|
|
525 |
\begin{textblock}{12}(1.7,12)
|
|
|
526 |
\only<3->{Backreferences: \bl{(\ldots)$\,\ldots{}\backslash{}n$}}
|
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|
527 |
\end{textblock}
|
|
|
528 |
\end{frame}
|
|
|
529 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
530 |
|
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|
531 |
|
|
|
532 |
|
|
|
533 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
534 |
\begin{frame}
|
|
|
535 |
\frametitle{Quiz 2}
|
|
|
536 |
|
|
|
537 |
\begin{textblock}{12}(2,2)
|
|
|
538 |
\onslide<1->{
|
|
|
539 |
\begin{bubble}[9.5cm]\it
|
|
|
540 |
Quiz:\smallskip\\
|
|
|
541 |
\textbf{What is the Thompson Construction for \bl{$r^{\{n\}}$} ?}
|
|
|
542 |
\end{bubble}}
|
|
|
543 |
\end{textblock}
|
|
|
544 |
|
|
|
545 |
\only<2->{
|
|
|
546 |
\begin{textblock}{12}(1,6)
|
|
|
547 |
\begin{tabular}{ll}
|
|
|
548 |
Google's RE2 lib & $\Rightarrow$ \bl{$a^{\{1001\}}$} is too big\\
|
|
|
549 |
\onslide<3->{Rust Regex lib} & \onslide<5->{$\Rightarrow$ \bl{$a^{\{1000\}\{100\}\{5\}}$} too big}\\
|
|
|
550 |
& \onslide<6->{$\Rightarrow$ \bl{$a^{\{0\}\{4294967295\}}$} ?}
|
|
|
551 |
\end{tabular}
|
|
|
552 |
\end{textblock}}
|
|
|
553 |
|
|
|
554 |
\only<4>{
|
|
|
555 |
\begin{textblock}{13.5}(1.5,4)
|
|
|
556 |
\begin{mybox3}{From Rust's Regex Description}\it
|
|
|
557 |
``\ldots [the] syntax is similar to Perl-style regular expressions, but lacks a few features like look
|
|
|
558 |
around and backreferences. In exchange, all searches execute in linear time with respect
|
|
|
559 |
to the size of the regular expression and search text. \ldots''
|
|
|
560 |
\end{mybox3}
|
|
|
561 |
\end{textblock}}
|
|
|
562 |
|
|
|
563 |
|
|
|
564 |
\end{frame}
|
|
876
|
565 |
|
|
|
566 |
|
|
|
567 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
913
|
568 |
\begin{frame}[t]
|
|
|
569 |
\frametitle{Regular Expressions}
|
|
876
|
570 |
|
|
913
|
571 |
\begin{textblock}{6}(2,5)
|
|
|
572 |
\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
|
|
|
573 |
\bl{$r$} & \bl{$::=$} & \bl{$\ZERO$} & nothing\\
|
|
|
574 |
& \bl{$\mid$} & \bl{$\ONE$} & empty string / \pcode{""} \\
|
|
|
575 |
& \bl{$\mid$} & \bl{$c, d, \ldots$} & characters\\
|
|
|
576 |
& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
|
|
|
577 |
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
|
|
|
578 |
& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
|
|
|
579 |
\end{tabular}
|
|
|
580 |
\end{textblock}
|
|
|
581 |
|
|
876
|
582 |
\end{frame}
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|
583 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
584 |
|
|
|
585 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
586 |
\begin{frame}[c]
|
|
913
|
587 |
\frametitle{\begin{tabular}{l}The Derivative of a Rexp\end{tabular}}
|
|
|
588 |
|
|
|
589 |
\large
|
|
|
590 |
If \bl{$r$} matches the string \bl{$c\!::\!s$}, what is a regular
|
|
|
591 |
expression that matches just \bl{$s$}?\bigskip\bigskip\bigskip\bigskip
|
|
|
592 |
|
|
|
593 |
\small
|
|
|
594 |
\bl{$\der\,c\,r$} gives the answer, Brzozowski 1964\medskip\\
|
|
|
595 |
(lost in the sands of time, re-appeared in 2009)
|
|
876
|
596 |
\end{frame}
|
|
|
597 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
598 |
|
|
|
599 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
600 |
\begin{frame}[c]
|
|
913
|
601 |
\frametitle{\begin{tabular}{l}The Derivative of a Rexp\end{tabular}}
|
|
876
|
602 |
|
|
913
|
603 |
\begin{center}
|
|
|
604 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
|
|
605 |
\bl{$\der\, c\, (\ZERO)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
|
|
|
606 |
\bl{$\der\, c\, (\ONE)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
|
|
|
607 |
\bl{$\der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
|
|
|
608 |
\bl{$\der\, c\, (r_1 + r_2)$} & \bl{$\dn$} & \bl{$\der\, c\, r_1 + \der\, c\, r_2$} & \\
|
|
|
609 |
\bl{$\der\, c\, (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{if $nullable (r_1)$}\\
|
|
|
610 |
& & \bl{then $(\der\,c\,r_1) \cdot r_2 + \der\, c\, r_2$}\\
|
|
|
611 |
& & \bl{else $(\der\, c\, r_1) \cdot r_2$}\\
|
|
|
612 |
\bl{$\der\, c\, (r^*)$} & \bl{$\dn$} & \bl{$(\der\,c\,r) \cdot (r^*)$} &\medskip\bigskip\\
|
|
|
613 |
\end{tabular}
|
|
|
614 |
\end{center}
|
|
876
|
615 |
|
|
|
616 |
\end{frame}
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|
|
617 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
618 |
|
|
|
619 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
913
|
620 |
\begin{frame}[t]
|
|
|
621 |
\frametitle{\begin{tabular}{l}Brzozowski matcher\end{tabular}}
|
|
876
|
622 |
|
|
913
|
623 |
Does \bl{$r_1$} match \bl{"$abc$"}?
|
|
|
624 |
\begin{center}
|
|
|
625 |
\begin{tabular}{@{}rl@{}l@{}}
|
|
|
626 |
Step 1: & build derivative of \bl{$a$} and \bl{$r_1$} & \bl{$(r_2 = \der\,a\,r_1)$}\smallskip\\
|
|
|
627 |
Step 2: & build derivative of \bl{$b$} and \bl{$r_2$} & \bl{$(r_3 = \der\,b\,r_2)$}\smallskip\\
|
|
|
628 |
Step 3: & build derivative of \bl{$c$} and \bl{$r_3$} & \bl{$(r_4 = \der\,c\,r_3)$}\smallskip\\
|
|
|
629 |
Step 4: & the string is exhausted: & \bl{($nullable(r_4))$}\\
|
|
|
630 |
& test whether \bl{$r_4$} can recognise\\
|
|
|
631 |
& the empty string\medskip\\
|
|
|
632 |
Output: & result of the test\\
|
|
|
633 |
& $\Rightarrow \bl{\textit{true}} \,\text{or}\,
|
|
|
634 |
\bl{\textit{false}}$\\
|
|
|
635 |
\end{tabular}
|
|
|
636 |
\end{center}
|
|
876
|
637 |
|
|
913
|
638 |
|
|
|
639 |
\end{frame}
|
|
|
640 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
641 |
|
|
|
642 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
643 |
\begin{frame}[c]
|
|
|
644 |
\frametitle{\mbox{Nullable}}
|
|
|
645 |
|
|
876
|
646 |
|
|
913
|
647 |
\ldots{}whether a regular expression can match the empty string:
|
|
|
648 |
\begin{center}
|
|
|
649 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
|
650 |
\bl{$nullable(\ZERO)$} & \bl{$\dn$} & \bl{\textit{false}}\\
|
|
|
651 |
\bl{$nullable(\ONE)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
|
|
652 |
\bl{$nullable (c)$} & \bl{$\dn$} & \bl{\textit{false}}\\
|
|
|
653 |
\bl{$nullable (r_1 + r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \vee nullable(r_2)$} \\
|
|
|
654 |
\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$} \\
|
|
|
655 |
\bl{$nullable (r^*)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
|
|
656 |
\end{tabular}
|
|
|
657 |
\end{center}
|
|
|
658 |
|
|
|
659 |
\end{frame}
|
|
|
660 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
661 |
|
|
|
662 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
663 |
\begin{frame}[t]
|
|
|
664 |
\frametitle{\begin{tabular}{l}Extensions\end{tabular}}
|
|
876
|
665 |
|
|
913
|
666 |
\begin{center}
|
|
|
667 |
\begin{tabular}{@ {\hspace{-4mm}}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
|
|
668 |
\bl{$\der\, c\, (r^{\{n\}})$} & \bl{$\dn$} & \bl{$if\;n=0\;then\;\ZERO\;else\; (der\,c\,r)\cdot r^{\{n-1\}}$} & \\
|
|
|
669 |
\bl{$\der\, c\, (r^{\{..n\}})$} & \bl{$\dn$} & \bl{$if\;n=0\;then\;\ZERO\;else\; (der\,c\,r)\cdot r^{\{n-1\}}$} & \\
|
|
|
670 |
\bl{$\der\, c\, (r^{\{n..\}})$} & \bl{$\dn$} & \bl{$if\;n=0\;then\;(der\,c\,r)\cdot r^*$} \\
|
|
|
671 |
& & \bl{$else\; (der\,c\,r)\cdot r^{\{n-1\}}$} & \\
|
|
|
672 |
\bl{$\der\, c\, (r^{\{n..m\}})$} & \bl{$\dn$} & \bl{$if\;n=0 \wedge m=0\;then\;\ZERO\;else$} & \\
|
|
|
673 |
& & \bl{$if \;n=0 \wedge 0<m\; then\;(der\,c\,r)\cdot r^{\{..m-1\}}$} & \\
|
|
|
674 |
& & \bl{$else\; (der\,c\,r)\cdot r^{\{n-1..m-1\}}$} & \\
|
|
|
675 |
\bl{$\der\, c\, (\neg{}r)$} & \bl{$\dn$} & \bl{$\neg(der\,c\,r)$}\\
|
|
|
676 |
\end{tabular}
|
|
|
677 |
\end{center}
|
|
|
678 |
|
|
|
679 |
also works for lookarounds, various anchors, etc, but \textbf{not} for backreferences
|
|
|
680 |
|
|
|
681 |
\begin{textblock}{3}(11,13)
|
|
|
682 |
\onslide<2>{
|
|
|
683 |
\begin{bubble}[3cm]
|
|
|
684 |
\bl{$a^{\{0\}\{4294967295\}}$}
|
|
|
685 |
\end{bubble}}
|
|
876
|
686 |
\end{textblock}
|
|
|
687 |
|
|
|
688 |
\end{frame}
|
|
|
689 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
690 |
|
|
|
691 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
913
|
692 |
\begin{frame}[t]
|
|
|
693 |
\frametitle{\mbox{Problems with Ders}}
|
|
876
|
694 |
|
|
913
|
695 |
\textcolor{blue}{
|
|
876
|
696 |
\small
|
|
913
|
697 |
\def\ll{\stackrel{der\,a}{\longrightarrow}}
|
|
876
|
698 |
\begin{center}
|
|
913
|
699 |
\begin{tabular}{@{\hspace{-5mm}}rll}
|
|
|
700 |
$(a + aa)^*$ & $\ll$ & $(\ONE + \ONE{}a) \cdot (a + aa)^*$\\
|
|
|
701 |
& $\ll$ & $(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* \;+\; (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
|
|
|
702 |
& $\ll$ & $(\ZERO + \ZERO{}a + \ZERO) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^* \;+\; $\\
|
|
|
703 |
& & $\qquad(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
|
|
|
704 |
& $\ll$ & \ldots\\ \hspace{15mm}
|
|
876
|
705 |
\end{tabular}
|
|
913
|
706 |
\end{center}}
|
|
|
707 |
|
|
|
708 |
\begin{textblock}{13.5}(1.5,12)
|
|
|
709 |
(regular expressions of sizes 98, 169, 283, 468, 767, \ldots)
|
|
|
710 |
\end{textblock}
|
|
876
|
711 |
|
|
|
712 |
\end{frame}
|
|
913
|
713 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
876
|
714 |
|
|
|
715 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
716 |
\begin{frame}[c]
|
|
913
|
717 |
\frametitle{\mbox{Conclusion}}
|
|
876
|
718 |
|
|
|
719 |
\begin{itemize}
|
|
913
|
720 |
\item The beauty is that this only involves functional programs that can be conveniently reasoned about in theorem provers
|
|
|
721 |
\item POSIX lexing can be done via an extension by Sulzmann and Lu, 2014 (our current work)\medskip\pause
|
|
|
722 |
\item How surprising that one can still do new work on regular expressions.
|
|
|
723 |
\end{itemize}
|
|
876
|
724 |
|
|
|
725 |
\end{frame}
|
|
|
726 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
727 |
|
|
|
728 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
729 |
|
|
913
|
730 |
\begin{frame}<1-10>
|
|
|
731 |
\end{frame}
|
|
876
|
732 |
|
|
|
733 |
|
|
|
734 |
|
|
|
735 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
736 |
\end{document}
|
|
|
737 |
|
|
|
738 |
%%% Local Variables:
|
|
|
739 |
%%% mode: latex
|
|
|
740 |
%%% TeX-master: t
|
|
|
741 |
%%% End:
|
|
|
742 |
|