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\documentclass[dvipsnames,14pt,t]{beamer}
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\usepackage{../slides}
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\usepackage{../graphics}
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\usepackage{../langs}
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\usepackage{../data}
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\pgfplotsset{compat=1.11}
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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% beamer stuff
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\renewcommand{\slidecaption}{CFL 03, King's College London}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm]
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\LARGE Compilers and \\[-2mm]
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\LARGE Formal Languages (3)\\[3mm]
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\end{tabular}}
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\normalsize
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\begin{center}
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\begin{tabular}{ll}
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Email: & christian.urban at kcl.ac.uk\\
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Office Hours: & Thursdays 12 -- 14\\
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Location: & N7.07 (North Wing, Bush House)\\
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Slides \& Progs: & KEATS (also homework is there)\\
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\end{tabular}
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Scala Book, Exams}
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\begin{itemize}
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\item \small https://nms.kcl.ac.uk/christian.urban/ProgInScala2ed.pdf\normalsize
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\item homework (written exam 80\%)
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\item coursework (20\%)\bigskip
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\item short survey at KEATS; to be answered until Sunday
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{Last Week}
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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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Last week I showed you a regular expression matcher
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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that works provably correct in all cases (we only
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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started with the proving part though)
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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{$matches\,s\,r$} \;\;if and only if \;\; \bl{$s \in L(r)$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}\bigskip\bigskip
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\small
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\textcolor{gray}{\mbox{}\hfill{}by Janusz Brzozowski (1964)}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{The Derivative of a Rexp}
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\begin{center}
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\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
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\bl{$der\, c\, (\ZERO)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
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\bl{$der\, c\, (\ONE)$} & \bl{$\dn$} & \bl{$\ZERO$} & \\
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\bl{$der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\, c\, (r_1 + r_2)$} & \bl{$\dn$} & \bl{$der\, c\, r_1 + der\, c\, r_2$} & \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\, c\, (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{if $nullable (r_1)$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& & \bl{then $(der\,c\,r_1) \cdot r_2 + der\, c\, r_2$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& & \bl{else $(der\, c\, r_1) \cdot r_2$}\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\, c\, (r^*)$} & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\medskip\\
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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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\bl{$\textit{ders}\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$\textit{ders}\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\textit{ders}\,s\,(der\,c\,r)$} & \\
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Example}
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Given \bl{$r \dn ((a \cdot b) + b)^*$} what is
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\begin{center}
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\def\arraystretch{1.5}
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\begin{tabular}{@{}lcl}
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\bl{$\der\,a\,((a \cdot b) + b)^*$}
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& \bl{$\Rightarrow$}& \bl{$\der\,a\, \underline{((a \cdot b) + b)^*}$}\\
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& \bl{$=$} & \bl{$(der\,a\,(\underline{(a \cdot b) + b})) \cdot r$}\\
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& \bl{$=$} & \bl{$((der\,a\,(\underline{a \cdot b})) + (der\,a\,b)) \cdot r$}\\
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& \bl{$=$} & \bl{$(((der\,a\,\underline{a}) \cdot b) + (der\,a\,b)) \cdot r$}\\
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& \bl{$=$} & \bl{$((\ONE \cdot b) + (der\,a\,\underline{b})) \cdot r$}\\
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& \bl{$=$} & \bl{$((\ONE \cdot b) + \ZERO) \cdot r$}\\
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\end{tabular}
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\end{center}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Input: string \bl{$abc$} and regular expression \bl{$r$}
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\begin{enumerate}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \bl{$der\,a\,r$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \bl{$der\,b\,(der\,a\,r)$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \bl{$der\,c\,(der\,b\,(der\,a\,r))$}\bigskip\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item finally check whether the last regular expression can match the empty string
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\end{enumerate}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Simplification}
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Given \bl{$r \dn ((a \cdot b) + b)^*$}, you can simplify as follows
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\begin{center}
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\def\arraystretch{2}
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\begin{tabular}{@{}lcl}
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\bl{$((\ONE \cdot b) + \ZERO) \cdot r$}
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& \bl{$\Rightarrow$} & \bl{$((\underline{\ONE \cdot b}) + \ZERO) \cdot r$}\\
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& \bl{$=$} & \bl{$(\underline{b + \ZERO}) \cdot r$}\\
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& \bl{$=$} & \bl{$b \cdot r$}\\
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\end{tabular}
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\end{center}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Proofs about Rexp}
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\begin{itemize}
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\item \bl{$P$} holds for \bl{$\ZERO$}, \bl{$\ONE$} and \bl{c}\bigskip
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\item \bl{$P$} holds for \bl{$r_1 + r_2$} under the assumption that \bl{$P$} already
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holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
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\item \bl{$P$} holds for \bl{$r_1 \cdot r_2$} under the assumption that \bl{$P$} already
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holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
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\item \bl{$P$} holds for \bl{$r^*$} under the assumption that \bl{$P$} already
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holds for \bl{$r$}.
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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We proved
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$nullable(r)$} \;if and only if\; \bl{$[] \in L(r)$}
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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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by induction on the regular expression \bl{$r$}.\bigskip\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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{\huge\bf\alert{Any Questions?}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{\begin{tabular}{c}Proofs about Natural\\[-1mm] Numbers and Strings\end{tabular}}
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\begin{itemize}
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\item \bl{$P$} holds for \bl{$0$} and
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\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} already
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holds for \bl{$n$}
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\end{itemize}\bigskip
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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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\item \bl{$P$} holds for \bl{$[]$} and
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\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} already
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holds for \bl{$s$}
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\end{itemize}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Correctness Proof \\[-1mm]
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for our Matcher}
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\begin{itemize}
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\item We started from
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\begin{center}
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\begin{tabular}{rp{4cm}}
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& \bl{$s \in L(r)$}\medskip\\
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\bl{$\Leftrightarrow$} & \bl{$[] \in \Ders\,s\,(L(r))$}\pause
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\end{tabular}
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\end{center}\bigskip
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\item \textbf{\alert{if}} we can show \bl{$\Ders\, s\,(L(r)) = L(\ders\,s\,r)$} we
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have\bigskip
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\begin{center}
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\begin{tabular}{rp{4cm}}
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\bl{$\Leftrightarrow$} & \bl{$[] \in L(\ders\,s\,r)$}\medskip\\
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\bl{$\Leftrightarrow$} & \bl{$nullable(\ders\,s\,r)$}\medskip\\
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\bl{$\dn$} & \bl{$matches\,s\,r$}
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\end{tabular}
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\end{center}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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We need to prove
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\begin{center}
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\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
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\end{center}
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also by induction on the regular expression \bl{$r$}.
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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{(Basic) Regular Expressions}
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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}{@ {}rrl@ {\hspace{13mm}}l}
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\bl{$r$} & \bl{$::=$} & \bl{$\ZERO$} & nothing\\
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& \bl{$\mid$} & \bl{$\ONE$} & empty string / "" / $[]$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& \bl{$\mid$} & \bl{$c$} & character\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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|
272 |
How about ranges \bl{$[a\mbox{-}z]$}, \bl{$r^+$} and \bl{$\sim{}r$}? Do they increase the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
273 |
set of languages we can recognise?
|
|
25
|
274 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
275 |
\end{frame}
|
|
25
|
276 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
277 |
|
|
|
278 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
279 |
\begin{frame}[c]
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
280 |
\frametitle{Negation}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
281 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
282 |
Assume you have an alphabet consisting of the letters \bl{$a$}, \bl{$b$} and \bl{$c$} only.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
283 |
Find a (basic!) regular expression that matches all strings \emph{\alert{except}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
284 |
\bl{$ab$} and \bl{$ac$}!
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
285 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
286 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
287 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
288 |
|
|
25
|
289 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
573
|
290 |
\begin{frame}[t]
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
291 |
\frametitle{Automata}
|
|
25
|
292 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
293 |
A \alert{\bf deterministic finite automaton}, DFA, consists of:
|
|
25
|
294 |
|
|
|
295 |
\begin{itemize}
|
|
515
|
296 |
\item an alphabet \bl{$\varSigma$}
|
|
574
|
297 |
\item a set of states \bl{$\mbox{Qs}$}
|
|
515
|
298 |
\item one of these states is the start state \bl{$\mbox{Q}_0$}
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
299 |
\item some states are accepting states \bl{$F$}, and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
300 |
\item there is transition function \bl{$\delta$}\bigskip
|
|
25
|
301 |
|
|
|
302 |
\small
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
303 |
which takes a state as argument and a character and produces a
|
|
515
|
304 |
new state; this function might not be everywhere defined $\Rightarrow$ partial function
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
305 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
306 |
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
307 |
\begin{center}
|
|
574
|
308 |
\bl{$A(\varSigma, \mbox{Qs}, \mbox{Q}_0, F, \delta)$}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
309 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
310 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
311 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
312 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
313 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
314 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
315 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
316 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
317 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
318 |
\begin{center}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
319 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
573
|
320 |
every state/.style={minimum size=0pt,inner sep=2pt,
|
|
|
321 |
draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
322 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
|
323 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
|
324 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
|
325 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
|
326 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
|
327 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
|
328 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
|
329 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
|
330 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
|
331 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
|
332 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
|
333 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
|
334 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
|
335 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
336 |
\end{tikzpicture}
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
337 |
\end{center}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
338 |
|
|
573
|
339 |
\mbox{}\\[-14mm]
|
|
|
340 |
\only<1>{
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
341 |
\begin{itemize}
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
342 |
\item the start state can be an accepting state
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
343 |
\item it is possible that there is no accepting state
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
344 |
\item all states might be accepting (but this does not necessarily mean all strings are accepted)
|
|
573
|
345 |
\end{itemize}}
|
|
|
346 |
\only<2>{
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
347 |
for this automaton \bl{$\delta$} is the function\\
|
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 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
350 |
\begin{tabular}{lll}
|
|
515
|
351 |
\bl{$(\mbox{Q}_0, a) \rightarrow \mbox{Q}_1$} & \bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_4$}
|
|
|
352 |
& \bl{$(\mbox{Q}_4, a) \rightarrow \mbox{Q}_4$}\\
|
|
|
353 |
\bl{$(\mbox{Q}_0, b) \rightarrow \mbox{Q}_2$} & \bl{$(\mbox{Q}_1, b) \rightarrow \mbox{Q}_2$} &
|
|
|
354 |
\bl{$(\mbox{Q}_4, b) \rightarrow \mbox{Q}_4$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
355 |
\end{tabular}\ldots
|
|
573
|
356 |
\end{center}
|
|
|
357 |
}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
358 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
359 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
360 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
361 |
|
|
573
|
362 |
|
135
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 |
\begin{frame}[t]
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
365 |
\frametitle{Accepting a String}
|
135
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 |
Given
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
368 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
369 |
\begin{center}
|
|
574
|
370 |
\bl{$A(\varSigma, \mbox{Qs}, \mbox{Q}_0, F, \delta)$}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
371 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
372 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
373 |
you can define
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
374 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
375 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
376 |
\begin{tabular}{l}
|
|
515
|
377 |
\bl{$\widehat{\delta}(q, []) \dn q$}\\
|
|
|
378 |
\bl{$\widehat{\delta}(q, c::s) \dn \widehat{\delta}(\delta(q, c), s)$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
379 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
380 |
\end{center}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
381 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
382 |
Whether a string \bl{$s$} is accepted by \bl{$A$}?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
383 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
384 |
\begin{center}
|
|
517
|
385 |
\hspace{5mm}\bl{$\widehat{\delta}(\mbox{Q}_0, s) \in F$}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
386 |
\end{center}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
387 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
388 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
389 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
390 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
391 |
\begin{frame}[c]
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
392 |
\frametitle{Regular Languages}
|
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 |
A \alert{\bf language} is a set of strings.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
395 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
396 |
A \alert{\bf regular expression} specifies a language.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
397 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
398 |
A language is \alert{\bf regular} iff there exists
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
399 |
a regular expression that recognises all its strings.\bigskip\bigskip\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
400 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
401 |
not all languages are regular, e.g.~\bl{$a^nb^n$} is not
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
402 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
403 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
404 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
405 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
406 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
407 |
\frametitle{Regular Languages (2)}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
408 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
409 |
A language is \alert{\bf regular} iff there exists a regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
410 |
expression that recognises all its strings.\bigskip\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
411 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
412 |
or {\bf equivalently}\bigskip\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
413 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
414 |
A language is \alert{\bf regular} iff there exists a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
415 |
deterministic finite automaton that recognises all its
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
416 |
strings.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
417 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
418 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
419 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
420 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
421 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
422 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
423 |
\begin{frame}[c]
|
348
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
424 |
\frametitle{\begin{tabular}{c}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
425 |
Non-Deterministic\\[-1mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
426 |
Finite Automata\end{tabular}}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
427 |
|
|
515
|
428 |
A non-deterministic finite automaton (NFA) consists again of:
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
429 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
430 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
431 |
\item a finite set of states
|
|
515
|
432 |
\item \underline{some} these states are the start states
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
433 |
\item some states are accepting states, and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
434 |
\item there is transition \alert{relation}\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
435 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
436 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
437 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
438 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
439 |
\begin{tabular}{c}
|
|
515
|
440 |
\bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_2$}\\
|
|
|
441 |
\bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_3$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
442 |
\end{tabular}
|
|
515
|
443 |
\bl{\ldots}
|
|
|
444 |
\hspace{10mm}\pause
|
|
|
445 |
\bl{$(\mbox{Q}_1, a) \rightarrow \{\mbox{Q}_2, \mbox{Q}_3\}$}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
446 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
447 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
448 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
449 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
450 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
451 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
452 |
\begin{frame}[c]
|
|
515
|
453 |
\frametitle{An NFA Example}
|
|
|
454 |
|
|
|
455 |
% A NFA for (ab* + b)*a
|
|
|
456 |
\begin{center}
|
|
|
457 |
\begin{tikzpicture}[>=stealth',very thick, auto,
|
|
|
458 |
every state/.style={minimum size=0pt,inner sep=3pt,
|
|
|
459 |
draw=blue!50,very thick,fill=blue!20},scale=2]
|
|
|
460 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
|
461 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
|
462 |
\node[state, accepting] (Q_2) [right=of Q_1] {$\mbox{Q}_2$};
|
|
|
463 |
\path[->] (Q_0) edge [loop above] node {\alert{$b$}} ();
|
|
|
464 |
\path[<-] (Q_0) edge node [below] {\alert{$b$}} (Q_1);
|
|
|
465 |
\path[->] (Q_0) edge [bend left] node [above] {\alert{$a$}} (Q_1);
|
|
|
466 |
\path[->] (Q_0) edge [bend right] node [below] {\alert{$a$}} (Q_2);
|
|
|
467 |
\path[->] (Q_1) edge [loop above] node {\alert{$a,b$}} ();
|
|
|
468 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_2);
|
|
|
469 |
\end{tikzpicture}
|
|
|
470 |
\end{center}
|
|
|
471 |
|
|
|
472 |
\end{frame}
|
|
|
473 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
474 |
|
|
|
475 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
476 |
\begin{frame}[c]
|
|
574
|
477 |
\frametitle{Another Example}
|
|
|
478 |
|
|
|
479 |
For the regular expression \bl{$(.^*)a\,(.^{\{n\}})bc$}\bigskip\bigskip
|
|
|
480 |
|
|
|
481 |
% \begin{center}
|
|
650
|
482 |
\mbox{}\hspace{-11.5mm}
|
|
574
|
483 |
\begin{tikzpicture}[>=stealth',very thick, auto,
|
|
|
484 |
every state/.style={minimum size=5pt,inner sep=1pt,
|
|
|
485 |
draw=blue!50,very thick,fill=blue!20},scale=1, node distance=5mm,
|
|
|
486 |
decoration={brace,mirror,amplitude=7}]
|
|
|
487 |
\node[state,initial] (Q_0) {$\;0\;$};
|
|
|
488 |
\node[state] (Q_1) [right=of Q_0] {$\;1\;$};
|
|
|
489 |
\node[state] (Q_2) [right=of Q_1] {$\;2\;$};
|
|
|
490 |
\node (Q_3) [right=of Q_2] {$\hspace{-1mm}\ldots\hspace{-3mm}$};
|
|
|
491 |
\node[state] (Q_4) [right=of Q_3] {$\;\phantom{1}\;$};
|
|
|
492 |
\node (Q_5) [right=of Q_4] {$\hspace{-1mm}\ldots\hspace{-3mm}$};
|
|
|
493 |
\node[state] (Q_6) [right=of Q_5] {\tiny$n+1$};
|
|
|
494 |
\node[state] (Q_7) [right=of Q_6] {\tiny$n+2$};
|
|
|
495 |
\node[state,accepting] (Q_8) [right=of Q_7] {\tiny$n+3$};
|
|
|
496 |
|
|
|
497 |
\path[->] (Q_0) edge [loop above] node {$*$} ();
|
|
|
498 |
\path[->] (Q_0) edge node [above] {$a$} (Q_1);
|
|
|
499 |
\path[->] (Q_1) edge node [above] {$*$} (Q_2);
|
|
|
500 |
\path[->] (Q_2) edge node [above] {$*$} (Q_3);
|
|
|
501 |
\path[->] (Q_4) edge node [above] {$*$} (Q_5);
|
|
|
502 |
\path[->] (Q_5) edge node [above] {$*$} (Q_6);
|
|
|
503 |
\path[->] (Q_6) edge node [above] {$b$} (Q_7);
|
|
|
504 |
\path[->] (Q_7) edge node [above] {$c$} (Q_8);
|
|
|
505 |
|
|
|
506 |
\draw [decorate] ([yshift=-5mm]Q_1.east) --node[below=3mm]{$n$} ([yshift=-5mm]Q_6.west);
|
|
|
507 |
\end{tikzpicture}\bigskip
|
|
|
508 |
|
|
|
509 |
|
|
|
510 |
{\small Note the star-transitions: accept any character.}
|
|
|
511 |
|
|
|
512 |
\end{frame}
|
|
|
513 |
|
|
|
514 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
515 |
\begin{frame}[c]
|
|
515
|
516 |
\frametitle{Two Epsilon NFA Examples}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
517 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
518 |
\small
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
519 |
\begin{center}
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
520 |
\begin{tabular}[t]{c@{\hspace{9mm}}c}
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
521 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
522 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
215
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
523 |
\node[state,initial] (q_0) {$q_0$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
524 |
\node[state] (q_1) [above=of q_0] {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
525 |
\node[state, accepting] (q_2) [below=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
526 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
527 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
528 |
\path[->] (q_0) edge [loop right] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
529 |
\path[->] (q_1) edge [loop above] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
530 |
\path[->] (q_2) edge [loop below] node {\alert{$b$}} ();
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
531 |
\end{tikzpicture} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
532 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
533 |
\raisebox{20mm}{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
534 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
535 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
536 |
\node[state,initial] (r_1) {$r_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
537 |
\node[state] (r_2) [above=of r_1] {$r_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
538 |
\node[state, accepting] (r_3) [right=of r_1] {$r_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
539 |
\path[->] (r_1) edge node [below] {\alert{$b$}} (r_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
540 |
\path[->] (r_2) edge [bend left] node [above] {\alert{$a$}} (r_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
541 |
\path[->] (r_1) edge [bend left] node [left] {\alert{$\epsilon$}} (r_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
542 |
\path[->] (r_2) edge [bend left] node [right] {\alert{$a$}} (r_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
543 |
\end{tikzpicture}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
544 |
\end{tabular}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
545 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
546 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
547 |
\end{frame}
|
135
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 |
|
|
515
|
550 |
|
135
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 |
\begin{frame}[c]
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
553 |
\frametitle{Rexp to NFA}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
554 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
555 |
\begin{center}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
556 |
\begin{tabular}[t]{l@{\hspace{10mm}}l}
|
|
445
|
557 |
\raisebox{1mm}{\bl{$\ZERO$}} &
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
558 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
559 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
560 |
\end{tikzpicture}\\\\
|
|
445
|
561 |
\raisebox{1mm}{\bl{$\ONE$}} &
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
562 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
563 |
\node[state, initial, accepting] (Q_0) {$\mbox{}$};
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
564 |
\end{tikzpicture}\\\\
|
|
651
|
565 |
\raisebox{5mm}{\bl{$c$}} &
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
566 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
567 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
|
568 |
\node[state, accepting] (Q_1) [right=of Q_0] {$\mbox{}$};
|
|
|
569 |
\path[->] (Q_0) edge node [below] {\alert{$c$}} (Q_1);
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
570 |
\end{tikzpicture}\\\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
571 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
572 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
573 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
574 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
575 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
576 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
577 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
578 |
\begin{frame}[t]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
579 |
\frametitle{Case $r_1\cdot r_2$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
580 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
581 |
\mbox{}\bigskip
|
|
515
|
582 |
\onslide<1->{By recursion we are given two automata:\bigskip}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
583 |
|
|
515
|
584 |
{\centering%
|
|
|
585 |
\only<1>{%
|
|
|
586 |
\begin{tikzpicture}[node distance=3mm,
|
|
|
587 |
>=stealth',very thick,
|
|
|
588 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20}]
|
|
|
589 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
|
590 |
\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
|
|
|
591 |
\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
|
|
|
592 |
\node (R_1) [right=of Q_0] {$\ldots$};
|
|
|
593 |
\node[state, accepting] (T_1) [right=of R_1] {$\mbox{}$};
|
|
|
594 |
\node[state, accepting] (T_2) [above=of T_1] {$\mbox{}$};
|
|
|
595 |
\node[state, accepting] (T_3) [below=of T_1] {$\mbox{}$};
|
|
|
596 |
|
|
|
597 |
\node (A_0) [right=2.5cm of T_1] {$\mbox{}$};
|
|
|
598 |
\node[state, initial] (A_01) [above=1mm of A_0] {$\mbox{}$};
|
|
|
599 |
\node[state, initial] (A_02) [below=1mm of A_0] {$\mbox{}$};
|
|
|
600 |
|
|
|
601 |
\node (b_1) [right=of A_0] {$\ldots$};
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
602 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
603 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
604 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
605 |
\begin{pgfonlayer}{background}
|
|
515
|
606 |
\node (1) [rounded corners, inner sep=1mm, thick,
|
|
|
607 |
draw=black!60, fill=black!20, fit= (Q_0) (R_1) (T_1) (T_2) (T_3)] {};
|
|
|
608 |
\node (2) [rounded corners, inner sep=1mm, thick,
|
|
|
609 |
draw=black!60, fill=black!20, fit= (A_0) (b_1) (c_1) (c_2) (c_3)] {};
|
|
|
610 |
\node [yshift=2mm] at (1.north) {$r_1$};
|
|
|
611 |
\node [yshift=2mm] at (2.north) {$r_2$};
|
143
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
612 |
\end{pgfonlayer}
|
|
515
|
613 |
\end{tikzpicture}}
|
|
|
614 |
\only<2>{%
|
|
|
615 |
\begin{tikzpicture}[node distance=3mm,
|
|
|
616 |
>=stealth',very thick,
|
|
|
617 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
|
|
618 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
|
619 |
\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
|
|
|
620 |
\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
|
|
|
621 |
\node (r_1) [right=of Q_0] {$\ldots$};
|
|
|
622 |
\node[state] (t_1) [right=of r_1] {$\mbox{}$};
|
|
|
623 |
\node[state] (t_2) [above=of t_1] {$\mbox{}$};
|
|
|
624 |
\node[state] (t_3) [below=of t_1] {$\mbox{}$};
|
|
|
625 |
|
|
|
626 |
\node (A_0) [right=2.5cm of t_1] {$\mbox{}$};
|
|
|
627 |
\node[state] (A_01) [above=1mm of A_0] {$\mbox{}$};
|
|
|
628 |
\node[state] (A_02) [below=1mm of A_0] {$\mbox{}$};
|
|
|
629 |
|
|
|
630 |
\node (b_1) [right=of A_0] {$\ldots$};
|
|
|
631 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
|
|
632 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
|
|
633 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
|
|
634 |
\path[->] (t_1) edge (A_01);
|
|
|
635 |
\path[->] (t_2) edge node [above] {$\epsilon$s} (A_01);
|
|
|
636 |
\path[->] (t_3) edge (A_01);
|
|
|
637 |
\path[->] (t_1) edge (A_02);
|
|
|
638 |
\path[->] (t_2) edge (A_02);
|
|
|
639 |
\path[->] (t_3) edge node [below] {$\epsilon$s} (A_02);
|
|
|
640 |
|
|
|
641 |
\begin{pgfonlayer}{background}
|
|
|
642 |
\node (3) [rounded corners, inner sep=1mm, thick,
|
|
|
643 |
draw=black!60, fill=black!20, fit= (Q_0) (c_1) (c_2) (c_3)] {};
|
|
|
644 |
\node [yshift=2mm] at (3.north) {$r_1\cdot r_2$};
|
|
|
645 |
\end{pgfonlayer}
|
|
|
646 |
\end{tikzpicture}}
|
|
|
647 |
%
|
|
|
648 |
}\bigskip\bigskip
|
137
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 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
651 |
We need to (1) change the accepting nodes of the first automaton into non-accepting nodes, and (2) connect them
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
652 |
via $\epsilon$-transitions to the starting state of the second automaton.
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
653 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
654 |
\end{frame}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
655 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
656 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
657 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
658 |
\begin{frame}[t]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
659 |
\frametitle{Case $r_1+ r_2$}
|
|
515
|
660 |
\mbox{}\\[-8mm]
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
661 |
|
|
515
|
662 |
\onslide<1->{By recursion we are given two automata:\smallskip}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
663 |
|
|
515
|
664 |
\hspace{2cm}
|
|
|
665 |
\only<1>{%
|
|
|
666 |
\begin{tikzpicture}[node distance=3mm,
|
|
|
667 |
>=stealth',very thick,
|
|
|
668 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
|
669 |
baseline=(current bounding box.center)]
|
|
|
670 |
\node at (0,0) (1) {$\mbox{}$};
|
|
|
671 |
\node (2) [above=14mm of 1] {};
|
|
|
672 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
|
673 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
|
674 |
\node[state, initial] (3) [below=10mm of 1] {$\mbox{}$};
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
675 |
|
|
515
|
676 |
\node (a) [right=of 2] {$\ldots\,$};
|
|
|
677 |
\node (a1) [right=of a] {$$};
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
678 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
679 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
680 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
681 |
\node (b) [right=of 3] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
682 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
683 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
684 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
685 |
\begin{pgfonlayer}{background}
|
|
515
|
686 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
|
|
687 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (3) (b1) (b2) (b3)] {};
|
|
|
688 |
\node [yshift=3mm] at (1.north) {$r_1$};
|
|
|
689 |
\node [yshift=3mm] at (2.north) {$r_2$};
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
690 |
\end{pgfonlayer}
|
|
515
|
691 |
\end{tikzpicture}}
|
|
|
692 |
\only<2>{%
|
|
|
693 |
\begin{tikzpicture}[node distance=3mm,
|
|
|
694 |
>=stealth',very thick,
|
|
|
695 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
|
696 |
baseline=(current bounding box.center)]
|
|
|
697 |
\node at (0,0) (1) {$\mbox{}$};
|
|
|
698 |
\node (2) [above=14mm of 1] {$$};
|
|
|
699 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
|
700 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
|
701 |
\node[state, initial] (3) [below=10mm of 1] {$\mbox{}$};
|
|
|
702 |
|
|
|
703 |
\node (a) [right=of 2] {$\ldots\,$};
|
|
|
704 |
\node (a1) [right=of a] {$$};
|
|
|
705 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
|
|
706 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
|
|
707 |
|
|
|
708 |
\node (b) [right=of 3] {$\ldots$};
|
|
|
709 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
|
|
710 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
|
|
711 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
|
|
712 |
|
|
|
713 |
%\path[->] (1) edge node [above] {$\epsilon$} (2);
|
|
|
714 |
%\path[->] (1) edge node [below] {$\epsilon$} (3);
|
|
|
715 |
|
|
|
716 |
\begin{pgfonlayer}{background}
|
|
|
717 |
\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3) (b2) (b3)] {};
|
|
|
718 |
\node [yshift=3mm] at (3.north) {$r_1+ r_2$};
|
|
|
719 |
\end{pgfonlayer}
|
|
|
720 |
\end{tikzpicture}}
|
|
|
721 |
%
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
722 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
723 |
\small
|
|
515
|
724 |
\mbox{}\\\medskip
|
|
517
|
725 |
We can just put both automata together.
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
726 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
727 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
728 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
729 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
730 |
\begin{frame}[c]
|
138
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
731 |
\frametitle{Case $r^*$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
732 |
|
|
515
|
733 |
\onslide<1->{By recursion we are given an automaton for $r$:\smallskip}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
734 |
|
|
515
|
735 |
\hspace{2cm}
|
|
|
736 |
\only<1>{\begin{tikzpicture}[node distance=3mm,
|
|
|
737 |
>=stealth',very thick,
|
|
|
738 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
|
739 |
baseline=(current bounding box.north)]
|
|
|
740 |
\node (2) {$\mbox{}$};
|
|
|
741 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
|
742 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
138
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
743 |
\node (a) [right=of 2] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
744 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
745 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
|
515
|
746 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
|
|
747 |
\begin{pgfonlayer}{background}
|
|
|
748 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
|
|
749 |
\node [yshift=3mm] at (1.north) {$r$};
|
|
|
750 |
\end{pgfonlayer}
|
|
|
751 |
\end{tikzpicture}}
|
|
|
752 |
\only<2->{\begin{tikzpicture}[node distance=3mm,
|
|
|
753 |
>=stealth',very thick,
|
|
|
754 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
|
755 |
baseline=(current bounding box.north)]
|
|
|
756 |
\node at (0,0) [state, initial,accepting] (1) {$\mbox{}$};
|
|
|
757 |
\node (2) [right=16mm of 1] {$\mbox{}$};
|
|
|
758 |
\node[state] (4) [above=1mm of 2] {$\mbox{}$};
|
|
|
759 |
\node[state] (5) [below=1mm of 2] {$\mbox{}$};
|
|
|
760 |
\node (a) [right=of 2] {$\ldots$};
|
138
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
761 |
\node[state] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
762 |
\node[state] (a2) [above=of a1] {$\mbox{}$};
|
|
515
|
763 |
\node[state] (a3) [below=of a1] {$\mbox{}$};
|
|
|
764 |
\path[->] (1) edge node [below] {$\epsilon$} (4);
|
|
|
765 |
\path[->] (1) edge node [below] {$\epsilon$} (5);
|
|
|
766 |
\path[->] (a1) edge [bend left=45] node [below] {$\epsilon$} (1);
|
|
|
767 |
\path[->] (a2) edge [bend right] node [below] {$\epsilon$} (1);
|
|
|
768 |
\path[->] (a3) edge [bend left=45] node [below] {$\epsilon$} (1);
|
138
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
769 |
\begin{pgfonlayer}{background}
|
|
515
|
770 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3)] {};
|
|
|
771 |
\node [yshift=3mm] at (2.north) {$r^*$};
|
138
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
772 |
\end{pgfonlayer}
|
|
515
|
773 |
\end{tikzpicture}}
|
|
|
774 |
\bigskip
|
138
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
775 |
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
776 |
\onslide<3->{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
777 |
Why can't we just have an epsilon transition from the accepting states to the starting state?}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
778 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
779 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
780 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
781 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
782 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
783 |
\begin{frame}[c]
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
784 |
\frametitle{Subset Construction}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
785 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
786 |
|
|
662
|
787 |
\begin{textblock}{5}(1,1.5)
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
788 |
\begin{center}
|
|
515
|
789 |
\begin{tikzpicture}[node distance=6mm,scale=0.7,>=stealth',very thick,
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
790 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
791 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
|
792 |
\node[state] (Q_1) [above=of Q_0] {$\mbox{Q}_1$};
|
|
|
793 |
\node[state, accepting] (Q_2) [below=of Q_0] {$\mbox{Q}_2$};
|
|
|
794 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_1);
|
|
|
795 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_2);
|
|
|
796 |
\path[->] (Q_0) edge [loop right] node {\alert{$a$}} ();
|
|
|
797 |
\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
|
798 |
\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
799 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
800 |
\end{center}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
801 |
\end{textblock}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
802 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
803 |
\begin{textblock}{11}(6.5,4.5)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
804 |
\begin{tabular}{r|cl}
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
805 |
nodes \textcolor{white}{*} & $a$ & $b$\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
806 |
\hline
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
807 |
$\{\}$ \textcolor{white}{*} & \onslide<2->{$\{\}$} & \onslide<2->{$\{\}$}\\
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
808 |
$\{0\}$ \textcolor{white}{*} & \onslide<3->{$\{0,1,2\}$} & \onslide<3->{$\{2\}$}\\
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
809 |
$\{1\}$ \textcolor{white}{*} & \onslide<3->{$\{1\}$} & \onslide<3->{$\{\}$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
810 |
$\{2\}$ \onslide<5->{*} & \onslide<3->{$\{\}$} & \onslide<3->{$\{2\}$}\\
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
811 |
$\{0,1\}$ \textcolor{white}{*} &\onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
812 |
$\{0,2\}$ \onslide<5->{*} &\onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
813 |
$\{1,2\}$ \onslide<5->{*} & \onslide<4->{$\{1\}$} & \onslide<4->{$\{2\}$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
814 |
\onslide<5->{s:} $\{0,1,2\}$ \onslide<5->{*}& \onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
815 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
816 |
\end{textblock}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
817 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
818 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
820 |
|
346
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 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
823 |
\frametitle{The Result}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
824 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
825 |
\footnotesize
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
826 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
827 |
\begin{tikzpicture}[scale=0.5,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
828 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
829 |
draw=blue!50,very thick,fill=blue!20},
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
830 |
baseline=0mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
831 |
\node[state,initial,accepting] (q012) {$\{0,1,2\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
832 |
\node[state,accepting] (q02) [right=of q012] {$\{0,2\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
833 |
\node[state] (q01) [above=of q02] {$\{0,1\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
834 |
\node[state,accepting] (q12) [below=of q02] {$\{1,2\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
835 |
\node[state] (q0) [right=2cm of q01] {$\{0\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
836 |
\node[state] (q1) [right=2.5cm of q02] {$\{1\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
837 |
\node[state,accepting] (q2) [right=1.5cm of q12] {$\{2\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
838 |
\node[state] (qn) [right=of q1] {$\{\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
839 |
|
348
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
840 |
\path[->] (q012) edge [loop below] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
841 |
\path[->] (q012) edge node [above] {\alert{$b$}} (q2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
842 |
\path[->] (q12) edge [bend left] node [below,pos=0.4] {\alert{$a$}} (q1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
843 |
\path[->] (q12) edge node [below] {\alert{$b$}} (q2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
844 |
\path[->] (q02) edge node [above] {\alert{$a$}} (q012);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
845 |
\path[->] (q02) edge [bend left] node [above, pos=0.8] {\alert{$b$}} (q2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
846 |
\path[->] (q01) edge node [below] {\alert{$a$}} (q012);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
847 |
\path[->] (q01) edge [bend left] node [above] {\alert{$b$}} (q2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
848 |
\path[->] (q0) edge node [below] {\alert{$a$}} (q012);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
849 |
\path[->] (q0) edge node [right, pos=0.2] {\alert{$b$}} (q2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
850 |
\path[->] (q1) edge [loop above] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
851 |
\path[->] (q1) edge node [above] {\alert{$b$}} (qn);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
852 |
\path[->] (q2) edge [loop right] node {\alert{$b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
853 |
\path[->] (q2) edge node [below] {\alert{$a$}} (qn);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
854 |
\path[->] (qn) edge [loop above] node {$\alert{a},\alert{b}$} ();
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
855 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
856 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
857 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
858 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
859 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
860 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
861 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
862 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
863 |
\begin{frame}[c]
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
864 |
\frametitle{Removing Dead States}
|
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 |
\footnotesize
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
867 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
868 |
\begin{tabular}{ll}
|
|
445
|
869 |
DFA: & (original) NFA:\\
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
870 |
\raisebox{10mm}{%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
871 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
872 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
873 |
draw=blue!50,very thick,fill=blue!20}]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
874 |
\node[state,initial,accepting] (q012) {$\{0,1,2\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
875 |
\node[state,accepting] (q2) [right=of q012] {$\{2\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
876 |
\node[state] (qn) [right=of q2] {$\{\}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
877 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
878 |
\path[->] (q012) edge [loop below] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
879 |
\path[->] (q012) edge node [above] {\alert{$b$}} (q2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
880 |
\path[->] (q2) edge [loop below] node {\alert{$b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
881 |
\path[->] (q2) edge node [below] {\alert{$a$}} (qn);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
882 |
\path[->] (qn) edge [loop above] node
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
883 |
{$\alert{a},\alert{b}$} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
884 |
\end{tikzpicture}}
|
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 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
887 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
888 |
draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
889 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
|
890 |
\node[state] (Q_1) [above=of Q_0] {$\mbox{Q}_1$};
|
|
|
891 |
\node[state, accepting] (Q_2) [below=of Q_0] {$\mbox{Q}_2$};
|
|
|
892 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_1);
|
|
|
893 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_2);
|
|
|
894 |
\path[->] (Q_0) edge [loop right] node {\alert{$a$}} ();
|
|
|
895 |
\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
|
896 |
\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
897 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
898 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
899 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
900 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
901 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
902 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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 |
|
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 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
907 |
\begin{frame}[c]
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
908 |
\frametitle{Regexps and Automata}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
909 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
910 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
911 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
912 |
\node (rexp) {\bl{\bf Regexps}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
913 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
914 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
915 |
\onslide<2->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
916 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
917 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
918 |
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
919 |
\end{tikzpicture}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
920 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
921 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
922 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
923 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
924 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
925 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
926 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
927 |
\frametitle{DFA Minimisation}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
928 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
929 |
\begin{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
930 |
\item Take all pairs \bl{$(q, p)$} with \bl{$q \not= p$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
931 |
\item Mark all pairs that accepting and non-accepting states
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
932 |
\item For all unmarked pairs \bl{$(q, p)$} and all characters \bl{$c$} test whether
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
933 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
934 |
\bl{$(\delta(q, c), \delta(p,c))$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
935 |
\end{center}
|
348
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
936 |
are marked. If yes in at least one case, then also mark \bl{$(q, p)$}.
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
937 |
\item Repeat last step until no change.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
938 |
\item All unmarked pairs can be merged.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
939 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
940 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
941 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
942 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
943 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
944 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
945 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
946 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
947 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
948 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
949 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
950 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
|
951 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
|
952 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
|
953 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
|
954 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
|
955 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
|
956 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
|
957 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
|
958 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
|
959 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
|
960 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
|
961 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
|
962 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
|
963 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
964 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
965 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
966 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
967 |
\mbox{}\\[-20mm]\mbox{}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
968 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
969 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
970 |
\begin{tikzpicture}[scale=0.8,line width=0.8mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
971 |
\draw (0,0) -- (4,0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
972 |
\draw (0,1) -- (4,1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
973 |
\draw (0,2) -- (3,2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
974 |
\draw (0,3) -- (2,3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
975 |
\draw (0,4) -- (1,4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
976 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
977 |
\draw (0,0) -- (0, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
978 |
\draw (1,0) -- (1, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
979 |
\draw (2,0) -- (2, 3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
980 |
\draw (3,0) -- (3, 2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
981 |
\draw (4,0) -- (4, 1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
982 |
|
|
515
|
983 |
\draw (0.5,-0.5) node {$\mbox{Q}_0$};
|
|
|
984 |
\draw (1.5,-0.5) node {$\mbox{Q}_1$};
|
|
|
985 |
\draw (2.5,-0.5) node {$\mbox{Q}_2$};
|
|
|
986 |
\draw (3.5,-0.5) node {$\mbox{Q}_3$};
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
987 |
|
|
515
|
988 |
\draw (-0.5, 3.5) node {$\mbox{Q}_1$};
|
|
|
989 |
\draw (-0.5, 2.5) node {$\mbox{Q}_2$};
|
|
|
990 |
\draw (-0.5, 1.5) node {$\mbox{Q}_3$};
|
|
|
991 |
\draw (-0.5, 0.5) node {$\mbox{Q}_4$};
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
992 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
993 |
\draw (0.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
994 |
\draw (1.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
995 |
\draw (2.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
996 |
\draw (3.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
997 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
998 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
999 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1000 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1001 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1002 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1003 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1004 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1005 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1006 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1007 |
\begin{tabular}{@{\hspace{-8mm}}cc@{}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1008 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1009 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
1010 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
|
1011 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
|
1012 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
|
1013 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
|
1014 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
|
1015 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
|
1016 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
|
1017 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
|
1018 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
|
1019 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
|
1020 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
|
1021 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
|
1022 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
|
1023 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1024 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1025 |
&
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1026 |
\raisebox{9mm}{\begin{tikzpicture}[scale=0.6,line width=0.8mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1027 |
\draw (0,0) -- (4,0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1028 |
\draw (0,1) -- (4,1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1029 |
\draw (0,2) -- (3,2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1030 |
\draw (0,3) -- (2,3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1031 |
\draw (0,4) -- (1,4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1032 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1033 |
\draw (0,0) -- (0, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1034 |
\draw (1,0) -- (1, 4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1035 |
\draw (2,0) -- (2, 3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1036 |
\draw (3,0) -- (3, 2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1037 |
\draw (4,0) -- (4, 1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1038 |
|
|
515
|
1039 |
\draw (0.5,-0.5) node {$\mbox{Q}_0$};
|
|
|
1040 |
\draw (1.5,-0.5) node {$\mbox{Q}_1$};
|
|
|
1041 |
\draw (2.5,-0.5) node {$\mbox{Q}_2$};
|
|
|
1042 |
\draw (3.5,-0.5) node {$\mbox{Q}_3$};
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1043 |
|
|
515
|
1044 |
\draw (-0.5, 3.5) node {$\mbox{Q}_1$};
|
|
|
1045 |
\draw (-0.5, 2.5) node {$\mbox{Q}_2$};
|
|
|
1046 |
\draw (-0.5, 1.5) node {$\mbox{Q}_3$};
|
|
|
1047 |
\draw (-0.5, 0.5) node {$\mbox{Q}_4$};
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1048 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1049 |
\draw (0.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1050 |
\draw (1.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1051 |
\draw (2.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1052 |
\draw (3.5,0.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1053 |
\draw (0.5,1.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1054 |
\draw (2.5,1.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1055 |
\draw (0.5,3.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1056 |
\draw (1.5,2.5) node {\large$\star$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1057 |
\end{tikzpicture}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1058 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1059 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1060 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1061 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1062 |
\mbox{}\\[-20mm]\mbox{}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1063 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1064 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1065 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1066 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
1067 |
\node[state,initial] (Q_02) {$\mbox{Q}_{0, 2}$};
|
|
|
1068 |
\node[state] (Q_13) [right=of Q_02] {$\mbox{Q}_{1, 3}$};
|
|
|
1069 |
\node[state, accepting] (Q_4) [right=of Q_13] {$\mbox{Q}_{4\phantom{,0}}$};
|
|
|
1070 |
\path[->] (Q_02) edge [bend left] node [above] {\alert{$a$}} (Q_13);
|
|
|
1071 |
\path[->] (Q_13) edge [bend left] node [below] {\alert{$b$}} (Q_02);
|
|
|
1072 |
\path[->] (Q_02) edge [loop below] node {\alert{$b$}} ();
|
|
|
1073 |
\path[->] (Q_13) edge node [above] {\alert{$a$}} (Q_4);
|
|
|
1074 |
\path[->] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1075 |
\end{tikzpicture}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1076 |
minimal automaton
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1077 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1078 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1079 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1080 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1081 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1082 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1083 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1084 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1085 |
\frametitle{Alternatives}
|
|
573
|
1086 |
\mbox{}\\[-14mm]\mbox{}
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1087 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1088 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1089 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1090 |
every state/.style={minimum size=0pt,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1091 |
inner sep=2pt,draw=blue!50,very thick,fill=blue!20}]
|
|
515
|
1092 |
\only<1>{\node[state,initial] (Q_0) {$\mbox{Q}_0$};}
|
|
|
1093 |
\only<2->{\node[state,accepting] (Q_0) {$\mbox{Q}_0$};}
|
|
|
1094 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
|
1095 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
|
1096 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
|
1097 |
\only<1>{\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};}
|
|
|
1098 |
\only<2->{\node[state, initial right] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};}
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1099 |
\only<1-2>{
|
|
515
|
1100 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
|
1101 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
|
1102 |
\path[->] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
|
1103 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
|
1104 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
|
1105 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
|
1106 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
|
1107 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
|
1108 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);}
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1109 |
\only<3->{
|
|
515
|
1110 |
\path[<-] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
|
1111 |
\path[<-] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
|
1112 |
\path[<-] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
|
1113 |
\path[<-] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
|
1114 |
\path[<-] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
|
1115 |
\path[<-] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
|
1116 |
\path[<-] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
|
1117 |
\path[<-] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
|
1118 |
\path[<-] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);}
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1119 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1120 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1121 |
\mbox{}\\[-18mm]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1122 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1123 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1124 |
\begin{itemize}
|
|
573
|
1125 |
\item<1-> exchange initial / accepting states\\[-2mm]
|
|
|
1126 |
\item<2-> reverse all edges\\[-2mm]
|
|
|
1127 |
\item<3-> subset construction $\Rightarrow$ DFA\\[-2mm]
|
|
|
1128 |
\item<4-> remove dead states\\[-2mm]
|
|
|
1129 |
\item<5-> repeat once more
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1130 |
\onslide<6->{$\Rightarrow$ minimal DFA}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1131 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1132 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1133 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1134 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1135 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1136 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1137 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1138 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1139 |
\frametitle{Regexps and Automata}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1140 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1141 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1142 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1143 |
\node (rexp) {\bl{\bf Regexps}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1144 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1145 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1146 |
\onslide<1->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1147 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1148 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1149 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1150 |
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1151 |
\end{tikzpicture}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1152 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1153 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1154 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1155 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1156 |
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1157 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1158 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1159 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1160 |
\begin{frame}<2>[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1161 |
\frametitle{DFA to Rexp}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1162 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1163 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1164 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1165 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
1166 |
\only<1>{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
|
|
1167 |
\only<2->{\node[state, initial,accepting] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
|
|
1168 |
\only<1>{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
|
1169 |
\only<2->{\node[state,accepting] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
|
1170 |
\only<1>{\node[state, accepting] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
|
1171 |
\only<2->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1172 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1173 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1174 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1175 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1176 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1177 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1178 |
;
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1179 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1180 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1181 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1182 |
\onslide<3>{How to get from a DFA to a regular expression?}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1183 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1184 |
\end{frame}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1185 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1186 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1187 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1188 |
\begin{frame}[c]
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1189 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1190 |
\begin{center}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1191 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1192 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
1193 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
|
|
1194 |
\only<1->{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
|
1195 |
\only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1196 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1197 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1198 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1199 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1200 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1201 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1202 |
;
|
139
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1203 |
\end{tikzpicture}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1204 |
\end{center}\pause\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1205 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1206 |
\onslide<2->{
|
|
445
|
1207 |
You know how to solve since school days, no?
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1208 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1209 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
|
515
|
1210 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$2\, \mbox{Q}_0 + 3 \,\mbox{Q}_1 + 4\, \mbox{Q}_2$}\\
|
|
|
1211 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$2 \,\mbox{Q}_0 + 3\, \mbox{Q}_1 + 1\, \mbox{Q}_2$}\\
|
|
|
1212 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$1\, \mbox{Q}_0 + 5\, \mbox{Q}_1 + 2\, \mbox{Q}_2$}\\
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1213 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1214 |
\end{tabular}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1215 |
\end{center}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1216 |
}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1217 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1218 |
\end{frame}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1219 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1220 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1221 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1222 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1223 |
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1224 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1225 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1226 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
515
|
1227 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
|
|
1228 |
\only<1->{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
|
1229 |
\only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1230 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1231 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1232 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1233 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1234 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1235 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1236 |
;
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1237 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1238 |
\end{center}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1239 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1240 |
\onslide<2->{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1241 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1242 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
|
650
|
1243 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,b + \mbox{Q}_1\,b + \mbox{Q}_2\,b + \ONE$}\\
|
|
515
|
1244 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a$}\\
|
|
|
1245 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_1\,a + \mbox{Q}_2\,a$}\\
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1246 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1247 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1248 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1249 |
}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1250 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1251 |
\onslide<3->{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1252 |
Arden's Lemma:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1253 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1254 |
If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1255 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1256 |
}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1257 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1258 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1259 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1260 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1261 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1262 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1263 |
\frametitle{Regexps and Automata}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1264 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1265 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1266 |
\begin{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1267 |
\node (rexp) {\bl{\bf Regexps}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1268 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1269 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1270 |
\onslide<1->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1271 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1272 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1273 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1274 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1275 |
\end{tikzpicture}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1276 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1277 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1278 |
\end{frame}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1279 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1280 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1281 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1282 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1283 |
\frametitle{Regular Languages (3)}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1284 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1285 |
A language is \alert{\bf regular} iff there exists a regular
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1286 |
expression that recognises all its strings.\bigskip\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1287 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1288 |
or {\bf equivalently}\bigskip\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1289 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1290 |
A language is \alert{\bf regular} iff there exists a
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1291 |
deterministic finite automaton that recognises all its
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1292 |
strings.\bigskip\medskip\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1293 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1294 |
Why is every finite set of strings a regular language?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1295 |
\end{frame}
|
265
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1296 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1297 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1298 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1299 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1300 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1301 |
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1302 |
Given the function
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1303 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1304 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1305 |
\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
|
|
445
|
1306 |
$rev(\ZERO)$ & $\dn$ & $\ZERO$\\
|
|
|
1307 |
$rev(\ONE)$ & $\dn$ & $\ONE$\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1308 |
$rev(c)$ & $\dn$ & $c$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1309 |
$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1310 |
$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1311 |
$rev(r^*)$ & $\dn$ & $rev(r)^*$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1312 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1313 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1314 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1315 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1316 |
and the set
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1317 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1318 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1319 |
\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1320 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1321 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1322 |
prove whether
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1323 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1324 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1325 |
\bl{$L(rev(r)) = Rev (L(r))$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1326 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1327 |
|
346
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1328 |
\end{frame}
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
1329 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
25
|
1330 |
|
|
|
1331 |
|
|
|
1332 |
\end{document}
|
|
|
1333 |
|
|
|
1334 |
%%% Local Variables:
|
|
|
1335 |
%%% mode: latex
|
|
|
1336 |
%%% TeX-master: t
|
|
|
1337 |
%%% End:
|
|
|
1338 |
|