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\documentclass[dvipsnames,14pt,t]{beamer}
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\usepackage{beamerthemeplainculight}
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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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\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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\lstset{language=Java,
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basicstyle=\ttfamily,
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keywordstyle=\color{javapurple}\bfseries,
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stringstyle=\color{javagreen},
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morecomment=[s][\color{javadocblue}]{/**}{*/},
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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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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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basicstyle=\ttfamily,
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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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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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% beamer stuff
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\renewcommand{\slidecaption}{AFL 02, King's College London, 3.~October 2012}
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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 (2)\\[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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\begin{tabular}{ll}
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Email: & christian.urban at kcl.ac.uk\\
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Of$\!$fice: & S1.27 (1st floor Strand Building)\\
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Slides: & KEATS
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\end{tabular}
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\end{center}
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\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}Languages\end{tabular}}
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A \alert{language} is a set of strings.\bigskip
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A \alert{regular expression} specifies a set of strings or language.
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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}[t]
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\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
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Their inductive definition:
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\begin{textblock}{6}(2,5)
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\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
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\bl{r} & \bl{$::=$} & \bl{$\varnothing$} & null\\
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& \bl{$\mid$} & \bl{$\epsilon$} & empty string / "" / []\\
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& \bl{$\mid$} & \bl{c} & character\\
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& \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$} & sequence\\
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& \bl{$\mid$} & \bl{r$_1$ + r$_2$} & alternative / choice\\
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& \bl{$\mid$} & \bl{r$^*$} & star (zero or more)\\
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\end{tabular}
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\end{textblock}
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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}[t]
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\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
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Their implementation in Scala:
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{\lstset{language=Scala}\fontsize{8}{10}\selectfont
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\texttt{\lstinputlisting{app51.scala}}}
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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 Meaning of a\\[-2mm] Regular Expression\end{tabular}}
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\begin{textblock}{15}(1,4)
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\begin{tabular}{@ {}rcl}
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\bl{$L$($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
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\bl{$L$($\epsilon$)} & \bl{$\dn$} & \bl{$\{$""$\}$}\\
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\bl{$L$(c)} & \bl{$\dn$} & \bl{$\{$"c"$\}$}\\
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\bl{$L$(r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{$L$(r$_1$) $\cup$ $L$(r$_2$)}\\
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\bl{$L$(r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{$L$(r$_1$) @ $L$(r$_2$)}\\
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\bl{$L$(r$^*$)} & \bl{$\dn$} & \bl{$\bigcup_{n \ge 0}$ $L$(r)$^n$}\\
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\end{tabular}\bigskip
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\hspace{5mm}\textcolor{gray}{$L$(r)$^0$ $\;\dn\;$ $\{$""$\}$}\\
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\textcolor{gray}{$L$(r)$^{n+1}$ $\;\dn\;$ $L$(r) @ $L$(r)$^n$}
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\end{textblock}
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\only<2->{
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\begin{textblock}{5}(11,5)
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\textcolor{gray}{\small
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A @ B\\
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\ldots you take out every string from A and
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concatenate it with every string in B
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}
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\end{textblock}}
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\only<3->{
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\begin{textblock}{6}(9,12)\small
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\bl{$L$} is a function from regular expressions to sets of strings\\
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\bl{$L$ : Rexp $\Rightarrow$ Set[String]}
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\end{textblock}}
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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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\large
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\begin{center}
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What is \bl{$L$(a$^*$)}?
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\YES}{\textcolor{gray}{yes}}
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\newcommand{\NO}{\textcolor{gray}{no}}
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\newcommand{\FORALLR}{\textcolor{gray}{$\forall$ r.}}
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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}Reg Exp Equivalences\end{tabular}}
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\begin{center}
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\begin{tabular}{l@ {\hspace{7mm}}rcl@ {\hspace{7mm}}l}
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&\bl{(a + b) + c} & \bl{$\equiv^?$} & \bl{a + (b + c)} & \onslide<2->{\YES}\\
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&\bl{a + a} & \bl{$\equiv^?$} & \bl{a} & \onslide<3->{\YES}\\
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&\bl{(a $\cdot$ b) $\cdot$ c} & \bl{$\equiv^?$} & \bl{a $\cdot$ (b $\cdot$ c)} & \onslide<4->{\YES}\\
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&\bl{a $\cdot$ a} & \bl{$\equiv^?$} & \bl{a} & \onslide<5->{\NO}\\
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&\bl{$\epsilon^*$} & \bl{$\equiv^?$} & \bl{$\epsilon$} & \onslide<6->{\YES}\\
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&\bl{$\varnothing^*$} & \bl{$\equiv^?$} & \bl{$\varnothing$} & \onslide<7->{\NO}\\
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\FORALLR &\bl{r $\cdot$ $\epsilon$} & \bl{$\equiv^?$} & \bl{r} & \onslide<8->{\YES}\\
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\FORALLR &\bl{r + $\epsilon$} & \bl{$\equiv^?$} & \bl{r} & \onslide<9->{\NO}\\
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\FORALLR &\bl{r + $\varnothing$} & \bl{$\equiv^?$} & \bl{r} & \onslide<10->{\YES}\\
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\FORALLR &\bl{r $\cdot$ $\varnothing$} & \bl{$\equiv^?$} & \bl{r} & \onslide<11->{\NO}\\
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&\bl{c $\cdot$ (a + b)} & \bl{$\equiv^?$} & \bl{(c $\cdot$ a) + (c $\cdot$ b)} & \onslide<12->{\YES}\\
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&\bl{a$^*$} & \bl{$\equiv^?$} & \bl{$\epsilon$ + (a $\cdot$ a$^*$)} & \onslide<13->{\YES}
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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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\frametitle{\begin{tabular}{c}The Meaning of Matching\end{tabular}}
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\large
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a regular expression \bl{r} matches a string \bl{s} is defined as
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\begin{center}
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\bl{s $\in$ $L$(r)}\\
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\end{center}\bigskip\bigskip\pause
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\small
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if \bl{r$_1$ $\equiv$ r$_2$}, then \bl{$s$ $\in$ $L$(r$_1$)} iff \bl{$s$ $\in$ $L$(r$_2$)}
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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}[t]
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\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}
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\begin{itemize}
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\item given a regular expression \bl{r} and a string \bl{s}, say yes or no for whether
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\begin{center}
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\bl{s $\in$ $L$(r)}
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\end{center}
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or not.\bigskip\bigskip\pause
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\end{itemize}\pause
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\small
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\begin{itemize}
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\item Identifiers (strings of letters or digits, starting with a letter)
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\item Integers (a non-empty sequence of digits)
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\item Keywords (else, if, while, \ldots)
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\item White space (a non-empty sequence of blanks, newlines and tabs)
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\end{itemize}
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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}A Matching Algorithm\end{tabular}}
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\small
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whether a regular expression matches the empty string:\medskip
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{\lstset{language=Scala}\fontsize{8}{10}\selectfont
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\texttt{\lstinputlisting{app5.scala}}}
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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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\large
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If \bl{r} matches the string \bl{c::s}, what is a regular expression that matches \bl{s}?\bigskip\bigskip\bigskip\bigskip
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\small
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\bl{der c r} gives the answer
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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 (2)\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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\bl{der c ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\
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\bl{der c ($\epsilon$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\
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\bl{der c (d)} & \bl{$\dn$} & \bl{if c $=$ d then $\epsilon$ else $\varnothing$} & \\
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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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\bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{if nullable r$_1$}\\
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& & \bl{then ((der c r$_1$) $\cdot$ r$_2$) + (der c r$_2$)}\\
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& & \bl{else (der c r$_1$) $\cdot$ r$_2$}\\
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\bl{der c (r$^*$)} & \bl{$\dn$} & \bl{(der c r) $\cdot$ (r$^*$)} &\smallskip\\\pause
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\bl{ders [] r} & \bl{$\dn$} & \bl{r} & \\
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\bl{ders (c::s) r} & \bl{$\dn$} & \bl{ders 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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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\begin{tabular}{c}The Derivative\end{tabular}}
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{\lstset{language=Scala}\fontsize{8}{10}\selectfont
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\texttt{\lstinputlisting{app6.scala}}}
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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 Rexp Matcher\end{tabular}}
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{\lstset{language=Scala}\fontsize{8}{10}\selectfont
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\texttt{\lstinputlisting{app7.scala}}}
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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}[t]
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\frametitle{\begin{tabular}{c}Proofs about Rexp\end{tabular}}
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Remember their inductive definition:\\[5cm]
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\begin{textblock}{6}(5,5)
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\begin{tabular}{@ {}rrl}
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\bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\
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& \bl{$\mid$} & \bl{$\epsilon$} \\
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& \bl{$\mid$} & \bl{c} \\
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& \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
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& \bl{$\mid$} & \bl{r$_1$ + r$_2$} \\
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& \bl{$\mid$} & \bl{r$^*$} \\
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\end{tabular}
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\end{textblock}
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If we want to prove something, say a property \bl{$P$(r)}, for all regular expressions \bl{r} then \ldots
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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}Proofs about Rexp (2)\end{tabular}}
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\begin{itemize}
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\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} 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$_2$}.
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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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\mode<presentation>{
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\begin{frame}[c]
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408 |
\frametitle{\begin{tabular}{c}Proofs about Rexp (3)\end{tabular}}
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409 |
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Assume \bl{$P(r)$} is the property:
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\begin{center}
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\bl{nullable(r)} if and only if \bl{"" $\in$ $L$(r)}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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419 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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420 |
\mode<presentation>{
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\begin{frame}[c]
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422 |
\frametitle{\begin{tabular}{c}Proofs about Strings\end{tabular}}
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423 |
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424 |
If we want to prove something, say a property \bl{$P$(s)}, for all strings \bl{s} then \ldots\bigskip
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425 |
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426 |
\begin{itemize}
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427 |
\item \bl{$P$} holds for the empty string, and\medskip
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|
428 |
\item \bl{$P$} holds for the string \bl{c::s} under the assumption that \bl{$P$}
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|
429 |
already holds for \bl{s}
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|
430 |
\end{itemize}
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431 |
\end{frame}}
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432 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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433 |
|
|
19
|
434 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
435 |
\mode<presentation>{
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|
436 |
\begin{frame}[c]
|
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|
437 |
\frametitle{\begin{tabular}{c}Proofs about Strings (2)\end{tabular}}
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438 |
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|
439 |
Let \bl{Der c A} be the set defined as
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440 |
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|
441 |
\begin{center}
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|
442 |
\bl{Der c A $\dn$ $\{$ s $|$ c::s $\in$ A$\}$ }
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|
443 |
\end{center}
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|
444 |
|
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|
445 |
Assume that \bl{$L$(der c r) = Der c ($L$(r))}. Prove that
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446 |
|
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|
447 |
\begin{center}
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|
448 |
\bl{matcher(r, s) if and only if s $\in$ $L$(r)}
|
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|
449 |
\end{center}
|
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|
450 |
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|
451 |
|
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|
452 |
\end{frame}}
|
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|
453 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
17
|
454 |
|
|
|
455 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
456 |
\mode<presentation>{
|
|
|
457 |
\begin{frame}[c]
|
|
|
458 |
\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}
|
|
|
459 |
|
|
|
460 |
A language (set of strings) is \alert{regular} iff there exists
|
|
|
461 |
a regular expression that recognises all its strings.
|
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|
462 |
|
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|
463 |
\end{frame}}
|
|
|
464 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
465 |
|
|
|
466 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
467 |
\mode<presentation>{
|
|
|
468 |
\begin{frame}[c]
|
|
|
469 |
\frametitle{\begin{tabular}{c}Automata\end{tabular}}
|
|
|
470 |
|
|
|
471 |
A deterministic finite automaton consists of:
|
|
|
472 |
|
|
|
473 |
\begin{itemize}
|
|
|
474 |
\item a set of states
|
|
|
475 |
\item one of these states is the start state
|
|
|
476 |
\item some states are accepting states, and
|
|
|
477 |
\item there is transition function\medskip
|
|
|
478 |
|
|
|
479 |
\small
|
|
|
480 |
which takes a state as argument and a character and produces a new state\smallskip\\
|
|
|
481 |
this function might not always be defined
|
|
|
482 |
\end{itemize}
|
|
|
483 |
|
|
|
484 |
\end{frame}}
|
|
|
485 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
486 |
|
|
7
|
487 |
|
|
|
488 |
\end{document}
|
|
|
489 |
|
|
|
490 |
%%% Local Variables:
|
|
|
491 |
%%% mode: latex
|
|
|
492 |
%%% TeX-master: t
|
|
|
493 |
%%% End:
|
|
|
494 |
|