\documentclass[dvipsnames,14pt,t]{beamer}\usepackage{beamerthemeplainculight}\usepackage[T1]{fontenc}\usepackage[latin1]{inputenc}\usepackage{mathpartir}\usepackage[absolute,overlay]{textpos}\usepackage{ifthen}\usepackage{tikz}\usepackage{pgf}\usepackage{calc} \usepackage{ulem}\usepackage{courier}\usepackage{listings}\renewcommand{\uline}[1]{#1}\usetikzlibrary{arrows}\usetikzlibrary{automata}\usetikzlibrary{shapes}\usetikzlibrary{shadows}\usetikzlibrary{positioning}\usetikzlibrary{calc}\usepackage{graphicx} \definecolor{javared}{rgb}{0.6,0,0} % for strings\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc\lstset{language=Java, basicstyle=\ttfamily, keywordstyle=\color{javapurple}\bfseries, stringstyle=\color{javagreen}, commentstyle=\color{javagreen}, morecomment=[s][\color{javadocblue}]{/**}{*/}, numbers=left, numberstyle=\tiny\color{black}, stepnumber=1, numbersep=10pt, tabsize=2, showspaces=false, showstringspaces=false}\lstdefinelanguage{scala}{ morekeywords={abstract,case,catch,class,def,% do,else,extends,false,final,finally,% for,if,implicit,import,match,mixin,% new,null,object,override,package,% private,protected,requires,return,sealed,% super,this,throw,trait,true,try,% type,val,var,while,with,yield}, otherkeywords={=>,<-,<\%,<:,>:,\#,@}, sensitive=true, morecomment=[l]{//}, morecomment=[n]{/*}{*/}, morestring=[b]", morestring=[b]', morestring=[b]"""}\lstset{language=Scala, basicstyle=\ttfamily, keywordstyle=\color{javapurple}\bfseries, stringstyle=\color{javagreen}, commentstyle=\color{javagreen}, morecomment=[s][\color{javadocblue}]{/**}{*/}, numbers=left, numberstyle=\tiny\color{black}, stepnumber=1, numbersep=10pt, tabsize=2, showspaces=false, showstringspaces=false}% beamer stuff \renewcommand{\slidecaption}{AFL 02, King's College London, 3.~October 2012}\newcommand{\bl}[1]{\textcolor{blue}{#1}} \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}<1>[t]\frametitle{% \begin{tabular}{@ {}c@ {}} \\[-3mm] \LARGE Automata and \\[-2mm] \LARGE Formal Languages (2)\\[3mm] \end{tabular}} %\begin{center} %\includegraphics[scale=0.3]{pics/ante1.jpg}\hspace{5mm} %\includegraphics[scale=0.31]{pics/ante2.jpg}\\ %\footnotesize\textcolor{gray}{Antikythera automaton, 100 BC (Archimedes?)} %\end{center}\normalsize \begin{center} \begin{tabular}{ll} Email: & christian.urban at kcl.ac.uk\\ Of$\!$fice: & S1.27 (1st floor Strand Building)\\ Slides: & KEATS \end{tabular} \end{center}\end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Languages\end{tabular}}A \alert{language} is a set of strings.\bigskipA \alert{regular expression} specifies a set of strings or language.\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[t]\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}Their inductive definition:\begin{textblock}{6}(2,5) \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l} \bl{r} & \bl{$::=$} & \bl{$\varnothing$} & null\\ & \bl{$\mid$} & \bl{$\epsilon$} & empty string / "" / []\\ & \bl{$\mid$} & \bl{c} & character\\ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$} & sequence\\ & \bl{$\mid$} & \bl{r$_1$ + r$_2$} & alternative / choice\\ & \bl{$\mid$} & \bl{r$^*$} & star (zero or more)\\ \end{tabular} \end{textblock}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[t]\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}Their implementation in Scala:{\lstset{language=Scala}\fontsize{8}{10}\selectfont\texttt{\lstinputlisting{app51.scala}}}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Meaning of a\\[-2mm] Regular Expression\end{tabular}}\begin{textblock}{15}(1,4) \begin{tabular}{@ {}rcl} \bl{$L$($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\ \bl{$L$($\epsilon$)} & \bl{$\dn$} & \bl{$\{$""$\}$}\\ \bl{$L$(c)} & \bl{$\dn$} & \bl{$\{$"c"$\}$}\\ \bl{$L$(r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{$L$(r$_1$) $\cup$ $L$(r$_2$)}\\ \bl{$L$(r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{$L$(r$_1$) @ $L$(r$_2$)}\\ \bl{$L$(r$^*$)} & \bl{$\dn$} & \bl{$\bigcup_{n \ge 0}$ $L$(r)$^n$}\\ \end{tabular}\bigskip\hspace{5mm}\textcolor{gray}{$L$(r)$^0$ $\;\dn\;$ $\{$""$\}$}\\\textcolor{gray}{$L$(r)$^{n+1}$ $\;\dn\;$ $L$(r) @ $L$(r)$^n$} \end{textblock}\only<2->{\begin{textblock}{5}(11,5)\textcolor{gray}{\smallA @ B\\\ldots you take out every string from A andconcatenate it with every string in B }\end{textblock}}\only<3->{\begin{textblock}{6}(9,12)\small\bl{$L$} is a function from regular expressions to sets of strings\\\bl{$L$ : Rexp $\Rightarrow$ Set[String]}\end{textblock}}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\large\begin{center}What is \bl{$L$(a$^*$)}?\end{center} \end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\YES}{\textcolor{gray}{yes}}\newcommand{\NO}{\textcolor{gray}{no}}\newcommand{\FORALLR}{\textcolor{gray}{$\forall$ r.}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Reg Exp Equivalences\end{tabular}}\begin{center}\begin{tabular}{l@ {\hspace{7mm}}rcl@ {\hspace{7mm}}l}&\bl{(a + b) + c} & \bl{$\equiv^?$} & \bl{a + (b + c)} & \onslide<2->{\YES}\\&\bl{a + a} & \bl{$\equiv^?$} & \bl{a} & \onslide<3->{\YES}\\&\bl{(a $\cdot$ b) $\cdot$ c} & \bl{$\equiv^?$} & \bl{a $\cdot$ (b $\cdot$ c)} & \onslide<4->{\YES}\\&\bl{a $\cdot$ a} & \bl{$\equiv^?$} & \bl{a} & \onslide<5->{\NO}\\&\bl{$\epsilon^*$} & \bl{$\equiv^?$} & \bl{$\epsilon$} & \onslide<6->{\YES}\\&\bl{$\varnothing^*$} & \bl{$\equiv^?$} & \bl{$\varnothing$} & \onslide<7->{\NO}\\\FORALLR &\bl{r $\cdot$ $\epsilon$} & \bl{$\equiv^?$} & \bl{r} & \onslide<8->{\YES}\\\FORALLR &\bl{r + $\epsilon$} & \bl{$\equiv^?$} & \bl{r} & \onslide<9->{\NO}\\\FORALLR &\bl{r + $\varnothing$} & \bl{$\equiv^?$} & \bl{r} & \onslide<10->{\YES}\\\FORALLR &\bl{r $\cdot$ $\varnothing$} & \bl{$\equiv^?$} & \bl{r} & \onslide<11->{\NO}\\&\bl{c $\cdot$ (a + b)} & \bl{$\equiv^?$} & \bl{(c $\cdot$ a) + (c $\cdot$ b)} & \onslide<12->{\YES}\\&\bl{a$^*$} & \bl{$\equiv^?$} & \bl{$\epsilon$ + (a $\cdot$ a$^*$)} & \onslide<13->{\YES}\end{tabular}\end{center}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Meaning of Matching\end{tabular}}\largea regular expression \bl{r} matches a string \bl{s} is defined as\begin{center}\bl{s $\in$ $L$(r)}\\ \end{center}\bigskip\bigskip\pause\smallif \bl{r$_1$ $\equiv$ r$_2$}, then \bl{$s$ $\in$ $L$(r$_1$)} iff \bl{$s$ $\in$ $L$(r$_2$)}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[t]\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}\begin{itemize}\item given a regular expression \bl{r} and a string \bl{s}, say yes or no for whether\begin{center}\bl{s $\in$ $L$(r)}\end{center}or not.\bigskip\bigskip\pause\end{itemize}\pause\small\begin{itemize}\item Identifiers (strings of letters or digits, starting with a letter)\item Integers (a non-empty sequence of digits)\item Keywords (else, if, while, \ldots)\item White space (a non-empty sequence of blanks, newlines and tabs)\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}\smallwhether a regular expression matches the empty string:\medskip{\lstset{language=Scala}\fontsize{8}{10}\selectfont\texttt{\lstinputlisting{app5.scala}}}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}\largeIf \bl{r} matches the string \bl{c::s}, what is a regular expression that matches \bl{s}?\bigskip\bigskip\bigskip\bigskip\small\bl{der c r} gives the answer\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Derivative of a Rexp (2)\end{tabular}}\begin{center}\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} \bl{der c ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\ \bl{der c ($\epsilon$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\ \bl{der c (d)} & \bl{$\dn$} & \bl{if c $=$ d then $\epsilon$ else $\varnothing$} & \\ \bl{der c (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{if nullable r$_1$}\\ & & \bl{then ((der c r$_1$) $\cdot$ r$_2$) + (der c r$_2$)}\\ & & \bl{else (der c r$_1$) $\cdot$ r$_2$}\\ \bl{der c (r$^*$)} & \bl{$\dn$} & \bl{(der c r) $\cdot$ (r$^*$)} &\smallskip\\\pause \bl{ders [] r} & \bl{$\dn$} & \bl{r} & \\ \bl{ders (c::s) r} & \bl{$\dn$} & \bl{ders s (der c r)} & \\ \end{tabular}\end{center}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Derivative\end{tabular}}{\lstset{language=Scala}\fontsize{8}{10}\selectfont\texttt{\lstinputlisting{app6.scala}}}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Rexp Matcher\end{tabular}}{\lstset{language=Scala}\fontsize{8}{10}\selectfont\texttt{\lstinputlisting{app7.scala}}}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[t]\frametitle{\begin{tabular}{c}Proofs about Rexp\end{tabular}}Remember their inductive definition:\\[5cm]\begin{textblock}{6}(5,5) \begin{tabular}{@ {}rrl} \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\ & \bl{$\mid$} & \bl{$\epsilon$} \\ & \bl{$\mid$} & \bl{c} \\ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\ & \bl{$\mid$} & \bl{r$_1$ + r$_2$} \\ & \bl{$\mid$} & \bl{r$^*$} \\ \end{tabular} \end{textblock}If we want to prove something, say a property \bl{$P$(r)}, for all regular expressions \bl{r} then \ldots\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Proofs about Rexp (2)\end{tabular}}\begin{itemize}\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} alreadyholds for \bl{r$_1$} and \bl{r$_2$}.\bigskip\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} alreadyholds for \bl{r$_1$} and \bl{r$_2$}.\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} alreadyholds for \bl{r}.\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Proofs about Rexp (3)\end{tabular}}Assume \bl{$P(r)$} is the property:\begin{center}\bl{nullable(r)} if and only if \bl{"" $\in$ $L$(r)}\end{center}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Proofs about Strings\end{tabular}}If we want to prove something, say a property \bl{$P$(s)}, for all strings \bl{s} then \ldots\bigskip\begin{itemize}\item \bl{$P$} holds for the empty string, and\medskip\item \bl{$P$} holds for the string \bl{c::s} under the assumption that \bl{$P$}already holds for \bl{s}\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Proofs about Strings (2)\end{tabular}}Let \bl{Der c A} be the set defined as\begin{center}\bl{Der c A $\dn$ $\{$ s $|$ c::s $\in$ A$\}$ } \end{center}Assume that \bl{$L$(der c r) = Der c ($L$(r))}. Prove that\begin{center}\bl{matcher(r, s) if and only if s $\in$ $L$(r)} \end{center}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}A language (set of strings) is \alert{regular} iff there existsa regular expression that recognises all its strings.\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Automata\end{tabular}}A deterministic finite automaton consists of:\begin{itemize}\item a set of states\item one of these states is the start state\item some states are accepting states, and\item there is transition function\medskip \smallwhich takes a state as argument and a character and produces a new state\smallskip\\this function might not always be defined\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}%%% Local Variables: %%% mode: latex%%% TeX-master: t%%% End: