\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 03, King's College London, 10.~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 (3)\\[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 (also home work is there)\\+
−
               & \alert{\bf (I have put a temporary link in there.)}\\+
−
  \end{tabular}+
−
  \end{center}+
−
+
−
+
−
\end{frame}}+
−
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}Last Week\end{tabular}}+
−
+
−
Last week I showed you+
−
+
−
\begin{itemize}+
−
\item one simple-minded regular expression matcher (which however does not work in all cases), and\bigskip+
−
\item one which works provably in all cases+
−
+
−
\begin{center}+
−
\bl{matcher r s} \;\;if and only if \;\; \bl{s $\in$ $L$(r)}+
−
\end{center} +
−
\end{itemize}+
−
  +
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}The Derivative of a Rexp\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$^*$)}\\+
−
  \end{tabular}+
−
\end{center}+
−
+
−
``the regular expression after \bl{c} has been recognised'' +
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
+
−
For this we defined the set \bl{Der c A} as+
−
+
−
\begin{center}+
−
\bl{Der c A $\dn$ $\{$ s $|$  c::s $\in$ A$\}$ } +
−
\end{center}+
−
+
−
which is called the semantic derivative of a set+
−
and proved +
−
+
−
\begin{center}+
−
\bl{$L$(der c r) $=$ Der c ($L$(r))}+
−
\end{center}+
−
+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}The Idea of the Algorithm\end{tabular}}+
−
+
−
If we want to recognise the string \bl{abc} with regular expression \bl{r}+
−
then\medskip+
−
+
−
\begin{enumerate}+
−
\item \bl{Der a ($L$(r))}\pause+
−
\item \bl{Der b (Der a ($L$(r)))}+
−
\item \bl{Der c (Der b (Der a ($L$(r))))}\pause+
−
\item finally we test whether the empty string is in set\pause\medskip+
−
\end{enumerate}+
−
+
−
The matching algorithm works similarly, just over regular expression than sets.+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
+
−
Input: string \bl{abc} and regular expression \bl{r} +
−
+
−
\begin{enumerate}+
−
\item \bl{der a r}+
−
\item \bl{der b (der a r)}+
−
\item \bl{der c (der b (der a r))}\pause+
−
\item finally check whether the latter regular expression can match the empty string+
−
\end{enumerate}+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
+
−
We need to prove+
−
+
−
\begin{center}+
−
\bl{$L$(der c r) $=$ Der c ($L$(r))}+
−
\end{center}+
−
+
−
by induction on the regular expression.+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}Proofs about Rexp\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{$P$} already+
−
holds 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{$P$} already+
−
holds for \bl{r$_1$} and \bl{r$_2$}.+
−
\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} already+
−
holds for \bl{r}.+
−
\end{itemize}+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}Proofs about Natural Numbers\\ and Strings\end{tabular}}+
−
+
−
\begin{itemize}+
−
\item \bl{$P$} holds for \bl{$0$} and+
−
\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} already+
−
holds for \bl{$n$}+
−
\end{itemize}\bigskip+
−
+
−
\begin{itemize}+
−
\item \bl{$P$} holds for \bl{\texttt{""}} and+
−
\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} already+
−
holds for \bl{$s$}+
−
\end{itemize}+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[t]+
−
\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}+
−
+
−
\begin{center}+
−
  \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}\bigskip\pause+
−
  \end{center}+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
+
−
+
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}Languages\end{tabular}}+
−
+
−
A \alert{language} is a set of strings.\bigskip+
−
+
−
A \alert{regular expression} specifies a set of strings or language.\bigskip+
−
+
−
A language is \alert{regular} iff there exists+
−
a regular expression that recognises all its strings.\bigskip\bigskip\pause+
−
+
−
\textcolor{gray}{not all languages are regular, e.g.~\bl{a$^n$b$^n$}.}+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[t]+
−
\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}+
−
+
−
\begin{center}+
−
  \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}\bigskip+
−
  \end{center}+
−
+
−
How about ranges \bl{[a-z]}, \bl{r$^\text{+}$} and \bl{!r}?+
−
  +
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}Negation of Regular Expr's\end{tabular}}+
−
+
−
\begin{itemize}+
−
\item \bl{!r}  \hspace{6mm} (everything that \bl{r} cannot recognise)\medskip+
−
\item \bl{$L$(!r) $\dn$ UNIV - $L$(r)}\medskip+
−
\item \bl{nullable (!r) $\dn$ not (nullable(r))}\medskip+
−
\item \bl{der\,c\,(!r) $\dn$ !(der\,c\,r)}+
−
\end{itemize}+
−
+
−
\end{frame}}+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
−
+
−
+
−
+
−
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
−
\mode<presentation>{+
−
\begin{frame}[c]+
−
\frametitle{\begin{tabular}{c}Regular Exp's for Lexing\end{tabular}}+
−
+
−
Lexing separates strings into ``words'' / components.+
−
+
−
\begin{itemize}+
−
\item Identifiers (non-empty strings of letters or digits, starting with a letter)+
−
\item Numbers (non-empty sequences of digits omitting leading zeros)+
−
\item Keywords (else, if, while, \ldots)+
−
\item White space (a non-empty sequence of blanks, newlines and tabs)+
−
\item Comments+
−
\end{itemize}+
−
+
−
\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 +
−
+
−
\small+
−
which 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: +
−
+
−