--- a/slides03.tex Sat Jun 15 09:11:11 2013 -0400
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,386 +0,0 @@
-\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}
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-\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}]{/**}{*/},
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- showstringspaces=false}
-
-\lstdefinelanguage{scala}{
- morekeywords={abstract,case,catch,class,def,%
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- for,if,implicit,import,match,mixin,%
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- morestring=[b]',
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-}
-
-\lstset{language=Scala,
- basicstyle=\ttfamily,
- keywordstyle=\color{javapurple}\bfseries,
- stringstyle=\color{javagreen},
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-
-% 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:
-