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\documentclass[dvipsnames,14pt,t]{beamer}
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\usepackage{beamerthemeplaincu}
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%%\usepackage[T1]{fontenc}
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\usepackage[latin1]{inputenc}
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\usepackage{mathpartir}
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\usepackage[absolute,overlay]{textpos}
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\usepackage{ifthen}
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\usepackage{tikz}
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\usepackage{pgf}
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\usepackage{calc}
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\usepackage{ulem}
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\usepackage{courier}
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\usepackage{listings}
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\renewcommand{\uline}[1]{#1}
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\usetikzlibrary{arrows}
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\usetikzlibrary{automata}
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\usetikzlibrary{shapes}
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\usetikzlibrary{shadows}
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\usetikzlibrary{positioning}
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\usetikzlibrary{calc}
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\usetikzlibrary{fit}
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\usetikzlibrary{backgrounds}
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\usepackage{graphicx}
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\definecolor{javared}{rgb}{0.6,0,0} % for strings
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\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
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\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
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\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\makeatletter
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\lst@CCPutMacro\lst@ProcessOther {"2D}{\lst@ttfamily{-{}}{-{}}}
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\@empty\z@\@empty
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\makeatother
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\lstset{language=Java,
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basicstyle=\consolas,
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keywordstyle=\color{javapurple}\bfseries,
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stringstyle=\color{javagreen},
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commentstyle=\color{javagreen},
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morecomment=[s][\color{javadocblue}]{/**}{*/},
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numbers=left,
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numberstyle=\tiny\color{black},
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stepnumber=1,
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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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\lstdefinelanguage{scala}{
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morekeywords={abstract,case,catch,class,def,%
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do,else,extends,false,final,finally,%
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for,if,implicit,import,match,mixin,%
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new,null,object,override,package,%
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private,protected,requires,return,sealed,%
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super,this,throw,trait,true,try,%
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type,val,var,while,with,yield},
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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otherkeywords={=>,<-,<\%,<:,>:,\#,@,->},
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sensitive=true,
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morecomment=[l]{//},
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morecomment=[n]{/*}{*/},
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morestring=[b]",
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morestring=[b]',
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morestring=[b]"""
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}
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\lstset{language=Scala,
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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basicstyle=\consolas,
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keywordstyle=\color{javapurple}\bfseries,
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stringstyle=\color{javagreen},
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commentstyle=\color{javagreen},
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morecomment=[s][\color{javadocblue}]{/**}{*/},
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numbers=left,
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numberstyle=\tiny\color{black},
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stepnumber=1,
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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% beamer stuff
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\slidecaption}{AFL 03, King's College London, 9.~October 2013}
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm]
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\LARGE Automata and \\[-2mm]
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\LARGE Formal Languages (3)\\[3mm]
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\end{tabular}}
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%\begin{center}
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%\includegraphics[scale=0.3]{pics/ante1.jpg}\hspace{5mm}
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%\includegraphics[scale=0.31]{pics/ante2.jpg}\\
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%\footnotesize\textcolor{gray}{Antikythera automaton, 100 BC (Archimedes?)}
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%\end{center}
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\normalsize
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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}{lp{8cm}}
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Email: & christian.urban at kcl.ac.uk\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Office: & S1.27 (1st floor Strand Building)\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Slides: & KEATS (also home work and coursework is there)\\
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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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\mode<presentation>{
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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}Regular Expressions\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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In programming languages they are often used to recognise:\medskip
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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 symbols, digits
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item identifiers
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item numbers (non-leading zeros)
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item keywords
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item comments
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{itemize}\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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\mbox{}\hfill\bl{\url{http://www.regexper.com}}
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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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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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\mode<presentation>{
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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}Last Week\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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which works provably correctly in all cases.
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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{$matcher\,r\,s$} \;\;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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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\, c\, (\varnothing)$} & \bl{$\dn$} & \bl{$\varnothing$} & \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\, c\, (\epsilon)$} & \bl{$\dn$} & \bl{$\varnothing$} & \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\epsilon$ else $\varnothing$} & \\
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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{$(der\,c\,r) \cdot (r^*)$} &\smallskip\\
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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{$der\!s\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$der\!s\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$der\!s\,s\,(der\,c\,r)$} & \\
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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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\mode<presentation>{
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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To see what is going on, define
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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{$Der\,c\,A \dn \{ s \;|\; c\!::\!s \in A\}$ }
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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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\small
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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For \bl{$A = \{"f\!oo", "bar", "f\!rak"\}$} then
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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{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$Der\,f\,A$ & $=$ & $\{"oo", "rak"\}$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$Der\,b\,A$ & $=$ & $\{"ar"\}$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$Der\,a\,A$ & $=$ & $\varnothing$\\
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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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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\begin{tabular}{c}The Idea of the Algorithm\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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If we want to recognise the string \bl{$"abc"$} with regular expression \bl{$r$}
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then\medskip
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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{$Der\,a\,(L(r))$}\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \bl{$Der\,b\,(Der\,a\,(L(r)))$}\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item \bl{$Der\,c\,(Der\,b\,(Der\,a\,(L(r))))$}\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\item finally we test whether the empty string is in this set\pause\medskip
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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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The matching algorithm works similarly, just over regular expression instead of sets.
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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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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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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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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We proved already
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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by induction on the regular expression.\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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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changeset
|
261 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
262 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
263 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
264 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
265 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
266 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
267 |
|
|
25
|
268 |
We need to prove
|
|
|
269 |
|
|
|
270 |
\begin{center}
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
271 |
\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
|
|
25
|
272 |
\end{center}
|
|
|
273 |
|
|
|
274 |
by induction on the regular expression.
|
|
|
275 |
\end{frame}}
|
|
|
276 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
277 |
|
|
|
278 |
|
|
|
279 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
280 |
\mode<presentation>{
|
|
|
281 |
\begin{frame}[c]
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
282 |
\frametitle{\begin{tabular}{c}Proofs about Rexps\end{tabular}}
|
|
25
|
283 |
|
|
|
284 |
\begin{itemize}
|
|
|
285 |
\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
286 |
\item \bl{$P$} holds for \bl{$r_1 + r_2$} under the assumption that \bl{$P$} already
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
287 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
288 |
\item \bl{$P$} holds for \bl{$r_1 \cdot r_2$} under the assumption that \bl{$P$} already
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
289 |
holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
290 |
\item \bl{$P$} holds for \bl{$r^*$} under the assumption that \bl{$P$} already
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
291 |
holds for \bl{$r$}.
|
|
25
|
292 |
\end{itemize}
|
|
|
293 |
|
|
|
294 |
\end{frame}}
|
|
|
295 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
296 |
|
|
|
297 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
298 |
\mode<presentation>{
|
|
|
299 |
\begin{frame}[c]
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
300 |
\frametitle{\begin{tabular}{c}Proofs about Natural\\[-1mm] Numbers and Strings\end{tabular}}
|
|
25
|
301 |
|
|
|
302 |
\begin{itemize}
|
|
|
303 |
\item \bl{$P$} holds for \bl{$0$} and
|
|
|
304 |
\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} already
|
|
|
305 |
holds for \bl{$n$}
|
|
|
306 |
\end{itemize}\bigskip
|
|
|
307 |
|
|
|
308 |
\begin{itemize}
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
309 |
\item \bl{$P$} holds for \bl{$""$} and
|
|
25
|
310 |
\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} already
|
|
|
311 |
holds for \bl{$s$}
|
|
|
312 |
\end{itemize}
|
|
|
313 |
|
|
|
314 |
\end{frame}}
|
|
|
315 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
316 |
|
|
|
317 |
|
|
|
318 |
|
|
|
319 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
320 |
\mode<presentation>{
|
|
|
321 |
\begin{frame}[c]
|
|
|
322 |
\frametitle{\begin{tabular}{c}Languages\end{tabular}}
|
|
|
323 |
|
|
|
324 |
A \alert{language} is a set of strings.\bigskip
|
|
|
325 |
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
326 |
A \alert{regular expression} specifies a language.\bigskip
|
|
25
|
327 |
|
|
|
328 |
A language is \alert{regular} iff there exists
|
|
|
329 |
a regular expression that recognises all its strings.\bigskip\bigskip\pause
|
|
|
330 |
|
134
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
331 |
\textcolor{gray}{not all languages are regular, e.g.~\bl{$a^nb^n$}.}
|
|
25
|
332 |
\end{frame}}
|
|
|
333 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
334 |
|
|
|
335 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
336 |
\mode<presentation>{
|
|
|
337 |
\begin{frame}[t]
|
|
|
338 |
\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
|
|
|
339 |
|
|
|
340 |
\begin{center}
|
134
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
341 |
\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
342 |
\bl{$r$} & \bl{$::=$} & \bl{$\varnothing$} & null\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
343 |
& \bl{$\mid$} & \bl{$\epsilon$} & empty string / "" / $[]$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
344 |
& \bl{$\mid$} & \bl{$c$} & character\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
345 |
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
346 |
& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
347 |
& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
348 |
\end{tabular}
|
|
25
|
349 |
\end{center}
|
134
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
350 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
351 |
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
|
352 |
set of languages we can recognise?
|
|
25
|
353 |
|
|
|
354 |
\end{frame}}
|
|
|
355 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
356 |
|
|
|
357 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
358 |
\mode<presentation>{
|
|
|
359 |
\begin{frame}[c]
|
|
|
360 |
\frametitle{\begin{tabular}{c}Negation of Regular Expr's\end{tabular}}
|
|
|
361 |
|
|
|
362 |
\begin{itemize}
|
134
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
363 |
\item \bl{$\sim{}r$} \hspace{6mm} (everything that \bl{r} cannot recognise)\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
364 |
\item \bl{$L(\sim{}r) \dn UNIV - L(r)$}\medskip
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
365 |
\item \bl{$nullable (\sim{}r) \dn not\, (nullable(r))$}\medskip
|
134
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
366 |
\item \bl{$der\,c\,(\sim{}r) \dn \;\sim{}(der\,c\,r)$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
367 |
\end{itemize}\bigskip\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
368 |
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
369 |
Used often for recognising comments:
|
134
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
370 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
371 |
\[
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
372 |
\bl{/ \cdot * \cdot (\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot * \cdot /}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
373 |
\]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
374 |
|
|
25
|
375 |
|
|
|
376 |
\end{frame}}
|
|
|
377 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
378 |
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
379 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
380 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
381 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
382 |
\frametitle{\begin{tabular}{c}Negation\end{tabular}}
|
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 |
Assume you have an alphabet consisting of the letters \bl{a}, \bl{b} and \bl{c} only.
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
385 |
Find a regular expression that matches all strings \emph{except} \bl{ab} and \bl{ac}.
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
386 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
387 |
\end{frame}}
|
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 |
|
|
25
|
390 |
|
|
|
391 |
|
|
|
392 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
393 |
\mode<presentation>{
|
|
|
394 |
\begin{frame}[c]
|
|
|
395 |
\frametitle{\begin{tabular}{c}Regular Exp's for Lexing\end{tabular}}
|
|
|
396 |
|
|
|
397 |
Lexing separates strings into ``words'' / components.
|
|
|
398 |
|
|
|
399 |
\begin{itemize}
|
|
|
400 |
\item Identifiers (non-empty strings of letters or digits, starting with a letter)
|
|
|
401 |
\item Numbers (non-empty sequences of digits omitting leading zeros)
|
|
|
402 |
\item Keywords (else, if, while, \ldots)
|
|
|
403 |
\item White space (a non-empty sequence of blanks, newlines and tabs)
|
|
|
404 |
\item Comments
|
|
|
405 |
\end{itemize}
|
|
|
406 |
|
|
|
407 |
\end{frame}}
|
|
|
408 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
409 |
|
|
|
410 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
411 |
\mode<presentation>{
|
|
|
412 |
\begin{frame}[c]
|
|
|
413 |
\frametitle{\begin{tabular}{c}Automata\end{tabular}}
|
|
|
414 |
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
415 |
A \alert{deterministic finite automaton} consists of:
|
|
25
|
416 |
|
|
|
417 |
\begin{itemize}
|
|
|
418 |
\item a set of states
|
|
|
419 |
\item one of these states is the start state
|
|
|
420 |
\item some states are accepting states, and
|
|
|
421 |
\item there is transition function\medskip
|
|
|
422 |
|
|
|
423 |
\small
|
|
|
424 |
which takes a state as argument and a character and produces a new state\smallskip\\
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
425 |
this function might not be everywhere defined
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
426 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
427 |
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
428 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
429 |
\bl{$A(Q, q_0, F, \delta)$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
430 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
431 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
432 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
433 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
434 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
435 |
|
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 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
438 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
439 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
440 |
\begin{center}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
441 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
442 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
443 |
\node[state,initial] (q_0) {$q_0$};
\node[state] (q_1) [right=of q_0] {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
444 |
\node[state] (q_2) [below right=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
445 |
\node[state] (q_3) [right=of q_2] {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
446 |
\node[state, accepting] (q_4) [right=of q_1] {$q_4$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
447 |
\path[->] (q_0) edge node [above] {\alert{$a$}} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
448 |
\path[->] (q_1) edge node [above] {\alert{$a$}} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
449 |
\path[->] (q_4) edge [loop right] node {\alert{$a, b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
450 |
\path[->] (q_3) edge node [right] {\alert{$a$}} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
451 |
\path[->] (q_2) edge node [above] {\alert{$a$}} (q_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
452 |
\path[->] (q_1) edge node [right] {\alert{$b$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
453 |
\path[->] (q_0) edge node [above] {\alert{$b$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
454 |
\path[->] (q_2) edge [loop left] node {\alert{$b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
455 |
\path[->] (q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (q_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
456 |
\end{tikzpicture}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
457 |
\end{center}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
458 |
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
459 |
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
460 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
461 |
\item start can be an accepting state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
462 |
\item it is possible that there is no accepting state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
463 |
\item all states might be accepting (but does not necessarily mean all strings are accepted)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
464 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
465 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
466 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
467 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
468 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
469 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
470 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
471 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
472 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
473 |
\begin{center}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
474 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
475 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
476 |
\node[state,initial] (q_0) {$q_0$};
\node[state] (q_1) [right=of q_0] {$q_1$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
477 |
\node[state] (q_2) [below right=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
478 |
\node[state] (q_3) [right=of q_2] {$q_3$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
479 |
\node[state, accepting] (q_4) [right=of q_1] {$q_4$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
480 |
\path[->] (q_0) edge node [above] {\alert{$a$}} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
481 |
\path[->] (q_1) edge node [above] {\alert{$a$}} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
482 |
\path[->] (q_4) edge [loop right] node {\alert{$a, b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
483 |
\path[->] (q_3) edge node [right] {\alert{$a$}} (q_4);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
484 |
\path[->] (q_2) edge node [above] {\alert{$a$}} (q_3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
485 |
\path[->] (q_1) edge node [right] {\alert{$b$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
486 |
\path[->] (q_0) edge node [above] {\alert{$b$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
487 |
\path[->] (q_2) edge [loop left] node {\alert{$b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
488 |
\path[->] (q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (q_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
489 |
\end{tikzpicture}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
490 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
491 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
492 |
for this automaton \bl{$\delta$} is the function\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
493 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
494 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
495 |
\begin{tabular}{lll}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
496 |
\bl{$(q_0, a) \rightarrow q_1$} & \bl{$(q_1, a) \rightarrow q_4$} & \bl{$(q_4, a) \rightarrow q_4$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
497 |
\bl{$(q_0, b) \rightarrow q_2$} & \bl{$(q_1, b) \rightarrow q_2$} & \bl{$(q_4, b) \rightarrow q_4$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
498 |
\end{tabular}\ldots
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
499 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
500 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
501 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
502 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
503 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
504 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
505 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
506 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
507 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
508 |
\begin{frame}[t]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
509 |
\frametitle{\begin{tabular}{c}Accepting a String\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
510 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
511 |
Given
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
512 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
513 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
514 |
\bl{$A(Q, q_0, F, \delta)$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
515 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
516 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
517 |
you can define
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
518 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
519 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
520 |
\begin{tabular}{l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
521 |
\bl{$\hat{\delta}(q, \texttt{""}) = q$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
522 |
\bl{$\hat{\delta}(q, c::s) = \hat{\delta}(\delta(q, c), s)$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
523 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
524 |
\end{center}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
525 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
526 |
Whether a string \bl{$s$} is accepted by \bl{$A$}?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
527 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
528 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
529 |
\hspace{5mm}\bl{$\hat{\delta}(q_0, s) \in F$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
530 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
531 |
\end{frame}}
|
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 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
534 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
535 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
536 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
537 |
\frametitle{\begin{tabular}{c}Non-Deterministic\\[-1mm] Finite Automata\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
538 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
539 |
A non-deterministic finite automaton consists again of:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
540 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
541 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
542 |
\item a finite set of states
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
543 |
\item one of these states is the start state
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
544 |
\item some states are accepting states, and
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
545 |
\item there is transition \alert{relation}\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
546 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
547 |
|
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 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
550 |
\begin{tabular}{c}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
551 |
\bl{$(q_1, a) \rightarrow q_2$}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
552 |
\bl{$(q_1, a) \rightarrow q_3$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
553 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
554 |
\hspace{10mm}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
555 |
\begin{tabular}{c}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
556 |
\bl{$(q_1, \epsilon) \rightarrow q_2$}\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
557 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
558 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
559 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
560 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
561 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
562 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
563 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
564 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
565 |
\begin{frame}[c]
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
566 |
\frametitle{An NFA Example}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
567 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
568 |
\begin{center}
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
569 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
570 |
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
|
571 |
\node[state,initial] (q_0) {$q_0$};
\node[state] (q_1) [above=of q_0] {$q_1$};
\node[state, accepting] (q_2) [below=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
572 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_1);
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
573 |
\path[->] (q_0) edge [loop right] node {\alert{$a$}} ();
\path[->] (q_1) edge [loop above] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
574 |
\path[->] (q_2) edge [loop below] node {\alert{$b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
575 |
\end{tikzpicture}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
576 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
577 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
578 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
579 |
\end{frame}}
|
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 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
582 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
583 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
584 |
\begin{frame}[c]
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
585 |
\frametitle{Rexp to NFA}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
586 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
587 |
\begin{center}
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
588 |
\begin{tabular}[t]{l@{\hspace{10mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
589 |
\raisebox{1mm}{\bl{$\varnothing$}} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
590 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
591 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
592 |
\end{tikzpicture}\\\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
593 |
\raisebox{1mm}{\bl{$\epsilon$}} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
594 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
595 |
\node[state, initial, accepting] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
596 |
\end{tikzpicture}\\\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
597 |
\raisebox{2mm}{\bl{$c$}} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
598 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
599 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
600 |
\node[state, accepting] (q_1) [right=of q_0] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
601 |
\path[->] (q_0) edge node [below] {\alert{$c$}} (q_1);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
602 |
\end{tikzpicture}\\\\
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
603 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
604 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
605 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
606 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
607 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
608 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
609 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
610 |
\mode<presentation>{
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
611 |
\begin{frame}[t]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
612 |
\frametitle{Case $r_1\cdot r_2$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
613 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
614 |
\mbox{}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
615 |
\onslide<1>{By recursion we are given two automata:\bigskip}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
616 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
617 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
618 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
619 |
\node[state, initial] (q_0) {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
620 |
\node (r_1) [right=of q_0] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
621 |
\only<1>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
622 |
\node[state, accepting] (t_1) [right=of r_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
623 |
\node[state, accepting] (t_2) [above=of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
624 |
\node[state, accepting] (t_3) [below=of t_1] {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
625 |
\only<2>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
626 |
\node[state] (t_1) [right=of r_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
627 |
\node[state] (t_2) [above=of t_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
628 |
\node[state] (t_3) [below=of t_1] {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
629 |
\only<1>{\node[state, initial] (a_0) [right=2.5cm of t_1] {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
630 |
\only<2>{\node[state] (a_0) [right=2.5cm of t_1] {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
631 |
\node (b_1) [right=of a_0] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
632 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
633 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
634 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
635 |
\only<2>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
636 |
\path[->] (t_1) edge node [above, pos=0.3] {\alert{$\epsilon$}} (a_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
637 |
\path[->] (t_2) edge node [above] {\alert{$\epsilon$}} (a_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
638 |
\path[->] (t_3) edge node [below] {\alert{$\epsilon$}} (a_0);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
639 |
}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
640 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
641 |
\only<1>{\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (q_0) (r_1) (t_1) (t_2) (t_3)] {};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
642 |
\only<1>{\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (a_0) (b_1) (c_1) (c_2) (c_3)] {};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
643 |
\only<2>{\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (q_0) (c_1) (c_2) (c_3)] {};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
644 |
\only<1>{\node [yshift=2mm] at (1.north) {\bl{$r_1$}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
645 |
\only<1>{\node [yshift=2mm] at (2.north) {\bl{$r_2$}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
646 |
\only<2>{\node [yshift=2mm] at (3.north) {\bl{$r_1\cdot r_2$}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
647 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
648 |
\end{tikzpicture}\bigskip\bigskip
|
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 |
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
654 |
\end{frame}}
|
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 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
659 |
\begin{frame}[t]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
660 |
\frametitle{Case $r_1+ r_2$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
661 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
662 |
\onslide<1>{By recursion we are given two automata:\smallskip}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
663 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
664 |
\begin{tikzpicture}[node distance=3mm,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
665 |
>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
666 |
\onslide<1>{\node at (0,0) (1) {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
667 |
\onslide<2>{\node at (0,0) [state, initial] (1) {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
668 |
\only<1>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
669 |
\node[state, initial] (2) [above right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
670 |
\node[state, initial] (3) [below right=16mm of 1] {$\mbox{}$};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
671 |
\only<2>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
672 |
\node[state] (2) [above right=16mm of 1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
673 |
\node[state] (3) [below right=16mm of 1] {$\mbox{}$};}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
674 |
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
675 |
\node (a) [right=of 2] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
676 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
677 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
678 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
679 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
680 |
\node (b) [right=of 3] {$\ldots$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
681 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
682 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
683 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
684 |
\only<2>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
685 |
\path[->] (1) edge node [above] {\alert{$\epsilon$}} (2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
686 |
\path[->] (1) edge node [below] {\alert{$\epsilon$}} (3);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
687 |
}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
688 |
\begin{pgfonlayer}{background}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
689 |
\only<1>{\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
690 |
\only<1>{\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (3) (b1) (b2) (b3)] {};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
691 |
\only<2>{\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3) (b2) (b3)] {};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
692 |
\only<1>{\node [yshift=3mm] at (1.north) {\bl{$r_1$}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
693 |
\only<1>{\node [yshift=3mm] at (2.north) {\bl{$r_2$}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
694 |
\only<2>{\node [yshift=3mm] at (3.north) {\bl{$r_1+ r_2$}};}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
695 |
\end{pgfonlayer}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
696 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
697 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
698 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
699 |
We (1) need to introduce a new starting state and (2) connect it to the original two starting states.
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
700 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
701 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
702 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
703 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
704 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
705 |
\begin{frame}[c]
|
137
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
706 |
\frametitle{Rexp to NFA}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
707 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
708 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
709 |
\begin{tabular}[b]{ll}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
710 |
\bl{r$^*$} & \includegraphics[scale=0.7]{pics/star.jpg}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
711 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
712 |
\end{center}\pause\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
713 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
714 |
Why can't we just have an epsilon transition from the accepting states to the starting state?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
715 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
716 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
717 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
718 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
719 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
720 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
721 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
722 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
723 |
\frametitle{\begin{tabular}{c}Subset Construction\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
724 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
725 |
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
726 |
\begin{textblock}{5}(1,1)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
727 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
728 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
729 |
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
|
730 |
\node[state,initial] (q_0) {$q_0$};
\node[state] (q_1) [above=of q_0] {$q_1$};
\node[state, accepting] (q_2) [below=of q_0] {$q_2$};
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
731 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_1);
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_2);
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
732 |
\path[->] (q_0) edge [loop right] node {\alert{$a$}} ();
\path[->] (q_1) edge [loop above] node {\alert{$a$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
733 |
\path[->] (q_2) edge [loop below] node {\alert{$b$}} ();
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
734 |
\end{tikzpicture}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
735 |
\end{center}
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
736 |
\end{textblock}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
737 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
738 |
\begin{textblock}{11}(6.5,4.5)
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
739 |
\begin{tabular}{r|cl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
740 |
& a & b\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
741 |
\hline
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
742 |
$\varnothing$ \onslide<2>{\textcolor{white}{*}} & $\varnothing$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
743 |
$\{0\}$ \onslide<2>{\textcolor{white}{*}} & $\{0,1,2\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
744 |
$\{1\}$ \onslide<2>{\textcolor{white}{*}} &$\{1\}$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
745 |
$\{2\}$ \onslide<2>{*} & $\varnothing$ &$\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
746 |
$\{0,1\}$ \onslide<2>{\textcolor{white}{*}} &$\{0,1,2\}$ &$\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
747 |
$\{0,2\}$ \onslide<2>{*}&$\{0,1,2\}$ &$\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
748 |
$\{1,2\}$ \onslide<2>{*}& $\{1\}$ & $\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
749 |
\onslide<2>{s:} $\{0,1,2\}$ \onslide<2>{*}&$\{0,1,2\}$ &$\{2\}$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
750 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
751 |
\end{textblock}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
752 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
753 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
754 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
755 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
756 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
757 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
758 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
759 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
760 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
761 |
\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
762 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
763 |
A language is \alert{regular} iff there exists
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
764 |
a regular expression that recognises all its strings.\bigskip\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
765 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
766 |
or equivalently\bigskip\medskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
767 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
768 |
A language is \alert{regular} iff there exists
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
769 |
a deterministic finite automaton that recognises all its strings.\bigskip\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
770 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
771 |
Why is every finite set of strings a regular language?
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
772 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
773 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
774 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
775 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
776 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
777 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
778 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
779 |
\begin{frame}[c]
|
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 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
782 |
\includegraphics[scale=0.5]{pics/ch3.jpg}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
783 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
784 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
785 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
786 |
\includegraphics[scale=0.5]{pics/ch4.jpg}\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
787 |
minimal automaton
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
788 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
789 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
790 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
791 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
792 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
793 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
794 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
795 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
796 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
797 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
798 |
\begin{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
799 |
\item Take all pairs \bl{(q, p)} with \bl{q $\not=$ p}
|
136
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
800 |
\item Mark all pairs that are accepting and non-accepting states
|
135
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
801 |
\item For all unmarked pairs \bl{(q, p)} and all characters \bl{c} tests wether
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
802 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
803 |
\bl{($\delta$(q,c), $\delta$(p,c))}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
804 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
805 |
are marked. If yes, then also mark \bl{(q, p)}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
806 |
\item Repeat last step until no chance.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
807 |
\item All unmarked pairs can be merged.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
808 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
809 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
810 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
811 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
812 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
813 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
814 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
815 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
816 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
817 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
818 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
819 |
Given the function
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
820 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
821 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
822 |
\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
823 |
$rev(\varnothing)$ & $\dn$ & $\varnothing$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
824 |
$rev(\epsilon)$ & $\dn$ & $\epsilon$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
825 |
$rev(c)$ & $\dn$ & $c$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
826 |
$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
827 |
$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
828 |
$rev(r^*)$ & $\dn$ & $rev(r)^*$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
829 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
830 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
831 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
832 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
833 |
and the set
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
834 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
835 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
836 |
\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
837 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
838 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
839 |
prove whether
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
840 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
841 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
842 |
\bl{$L(rev(r)) = Rev (L(r))$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
843 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
844 |
|
133
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
845 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
846 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
25
|
847 |
|
|
|
848 |
|
|
|
849 |
\end{document}
|
|
|
850 |
|
|
|
851 |
%%% Local Variables:
|
|
|
852 |
%%% mode: latex
|
|
|
853 |
%%% TeX-master: t
|
|
|
854 |
%%% End:
|
|
|
855 |
|