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%%\usepackage[T1]{fontenc}
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\usepackage{mathpartir}
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\usepackage{ifthen}
\usepackage{tikz}
\usepackage{pgf}
\usepackage{calc}
\usepackage{ulem}
\usepackage{courier}
\usepackage{listings}
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basicstyle=\consolas,
keywordstyle=\color{javapurple}\bfseries,
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morecomment=[s][\color{javadocblue}]{/**}{*/},
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% beamer stuff
\renewcommand{\slidecaption}{AFL 02, King's College London, 2.~October 2013}
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\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\\
Office: & 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.\bigskip
A \alert{regular expression} specifies a set of strings, or language.
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}Strings\end{tabular}}
Different ways of writing strings:
\begin{center}
\begin{tabular}{ccc}
\bl{\consolas"$hello$"}\quad{} & \bl{$[h, e, l, l, o]$} & \quad\bl{$h\!::\!e\!::\!l\!::\!l\!::\!o\!::\!N\!il$}\bigskip\\
\bl{\consolas ""} & \bl{$[]$} & \bl{$N$\!$il$}
\end{tabular}
\end{center}\pause
The concatenation operation on strings and sets of strings:
\begin{center}
\begin{tabular}{l}
\bl{{\consolas"$f\!oo$"}$\;@\;${\consolas"$bar$"}$\;=\;${\consolas"$f\!oobar$"}}\medskip\\
\bl{$A \;@\; B \dn \{ s_1 @ s_2 \mid s_1 \in A \wedge s_2 \in B\}$}
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
Their inductive definition:
\begin{textblock}{6}(2,6.5)
\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
\bl{$r$} & \bl{$::=$} & \bl{$\varnothing$} & null\\
& \bl{$\mid$} & \bl{$\epsilon$} & empty string / {\consolas""} / $[]$\\
& \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}
\only<2->{
\begin{textblock}{9}(4,0.5)
\begin{tikzpicture}
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\normalsize\color{darkgray}
\begin{minipage}{9cm}
\hspace{5mm}\mbox{{\lstset{language=Scala}\fontsize{8}{10}\selectfont
\texttt{\lstinputlisting{../progs/app51.scala}}}}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\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
\only<2->{
\hspace{5mm}\textcolor{blue}{$L(r)^0 \;\dn\; \{""\}$}\\
\textcolor{blue}{$L(r)^{n+1} \;\dn\; L(r) \,@\, L(r)^n$}}
\end{textblock}
\only<1->{
\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 Specification\\[-1mm] for Matching\end{tabular}}
\large
a regular expression \bl{$r$} matches a string \bl{$s$}\\ if and only if
\begin{center}
\bl{$s \in L(r)$}\\
\end{center}\bigskip\bigskip\pause
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}
\mbox{}\\[-13mm]
\begin{tikzpicture}[y=.2cm, x=.3cm]
%axis
\draw (0,0) -- coordinate (x axis mid) (30,0);
\draw (0,0) -- coordinate (y axis mid) (0,30);
%ticks
\foreach \x in {0,5,...,30}
\draw (\x,1pt) -- (\x,-3pt)
node[anchor=north] {\x};
\foreach \y in {0,5,...,30}
\draw (1pt,\y) -- (-3pt,\y)
node[anchor=east] {\y};
%labels
\node[below=0.6cm] at (x axis mid) {\bl{a}s};
\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
%plots
\draw[color=blue] plot[mark=*, mark options={fill=white}]
file {re-python.data};
\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ]
file {re-ruby.data};
%legend
\begin{scope}[shift={(4,20)}]
\draw[color=blue] (0,0) --
plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Python};
\draw[yshift=-\baselineskip, color=brown] (0,0) --
plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Ruby};
\end{scope}
\end{tikzpicture}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Evil Regular Expressions\end{tabular}}
\begin{itemize}
\item \alert{R}egular \alert{e}xpression \alert{D}enial \alert{o}f \alert{S}ervice (ReDoS)\bigskip
\item Evil regular expressions\medskip
\begin{itemize}
\item \bl{$(a?\{n\}) \cdot a\{n\}$}
\item \bl{$(a^+)^+$}
\item \bl{$([a$\,-\,$z]^+)^*$}
\item \bl{$(a + a \cdot a)^+$}
\item \bl{$(a + a?)^+$}
\end{itemize}
\end{itemize}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}
\small
\ldots{}whether a regular expression can match the empty string:
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
\bl{$nullable(\varnothing)$} & \bl{$\dn$} & \bl{$f\!\/alse$}\\
\bl{$nullable(\epsilon)$} & \bl{$\dn$} & \bl{$true$}\\
\bl{$nullable (c)$} & \bl{$\dn$} & \bl{$f\!alse$}\\
\bl{$nullable (r_1 + r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \vee nullable(r_2)$} \\
\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$} \\
\bl{$nullable (r^*)$} & \bl{$\dn$} & \bl{$true$} \\
\end{tabular}
\end{center}
\only<2->{
\begin{textblock}{9}(3.4,10)
\begin{tikzpicture}
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\normalsize\color{darkgray}
\begin{minipage}{9cm}
\hspace{5mm}\mbox{{\lstset{language=Scala}\fontsize{8}{10}\selectfont
\texttt{\lstinputlisting{../progs/app5.scala}}}}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}
\large
If \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{$der\!s\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
\bl{$der\!s\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$der\!s\,s\,(der\,c\,r)$} & \\
\end{tabular}
\end{center}
\only<3->{
\begin{textblock}{10.5}(2,5)
\begin{tikzpicture}
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
{\normalsize\color{darkgray}
\begin{minipage}{10.5cm}
\hspace{5mm}\mbox{{\lstset{language=Scala}\fontsize{8}{10}\selectfont
\texttt{\lstinputlisting{../progs/app6.scala}}}}
\end{minipage}};
\end{tikzpicture}
\end{textblock}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
Given \bl{$r \dn ((a \cdot b) + b)^*$} what is
\begin{center}
\begin{tabular}{l}
\bl{$der\,a\,r$}\\
\bl{$der\,b\,r$}
\end{tabular}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}
\mbox{}\\[-13mm]
\begin{tikzpicture}[y=.2cm, x=.3cm]
%axis
\draw (0,0) -- coordinate (x axis mid) (30,0);
\draw (0,0) -- coordinate (y axis mid) (0,30);
%ticks
\foreach \x in {0,5,...,30}
\draw (\x,1pt) -- (\x,-3pt)
node[anchor=north] {\x};
\foreach \y in {0,5,...,30}
\draw (1pt,\y) -- (-3pt,\y)
node[anchor=east] {\y};
%labels
\node[below=0.6cm] at (x axis mid) {\bl{a}s};
\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
%plots
\draw[color=blue] plot[mark=*, mark options={fill=white}]
file {re-python.data};
\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
file {re1.data};
\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ]
file {re-ruby.data};
%legend
\begin{scope}[shift={(4,20)}]
\draw[color=blue] (0,0) --
plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Python};
\draw[yshift=-\baselineskip, color=brown] (0,0) --
plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Ruby};
\draw[yshift=\baselineskip, color=red] (0,0) --
plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Scala V1};
\end{scope}
\end{tikzpicture}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}Proofs about Rexps\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{$P$} 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{$r_2$}.\bigskip
\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 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 Rexp (4)\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}
We can prove
\begin{center}
\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
\end{center}
by induction on \bl{$r$}.
\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}}
We can finally prove
\begin{center}
\bl{$matcher(r, s)$} if and only if \bl{$s \in L(r)$}
\end{center}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}
\mbox{}\\[-13mm]
\begin{tikzpicture}[y=.2cm, x=.3cm]
%axis
\draw (0,0) -- coordinate (x axis mid) (30,0);
\draw (0,0) -- coordinate (y axis mid) (0,30);
%ticks
\foreach \x in {0,5,...,30}
\draw (\x,1pt) -- (\x,-3pt)
node[anchor=north] {\x};
\foreach \y in {0,5,...,30}
\draw (1pt,\y) -- (-3pt,\y)
node[anchor=east] {\y};
%labels
\node[below=0.6cm] at (x axis mid) {\bl{a}s};
\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
%plots
\draw[color=blue] plot[mark=*, mark options={fill=white}]
file {re-python.data};
\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
file {re1.data};
\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ]
file {re-ruby.data};
%legend
\begin{scope}[shift={(4,20)}]
\draw[color=blue] (0,0) --
plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Python};
\draw[yshift=-\baselineskip, color=brown] (0,0) --
plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Ruby};
\draw[yshift=\baselineskip, color=red] (0,0) --
plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Scala V1};
\end{scope}
\end{tikzpicture}
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}A Problem\end{tabular}}
We represented the ``n-times'' \bl{$a\{n\}$} as a sequence regular expression:
\begin{center}
\begin{tabular}{rl}
1: & \bl{$a$}\\
2: & \bl{$a\cdot a$}\\
3: & \bl{$a\cdot a\cdot a$}\\
& \ldots\\
13: & \bl{$a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a$}\\
& \ldots\\
20:
\end{tabular}
\end{center}
This problem is aggravated with \bl{$a?$} being represented as \bl{$\epsilon + a$}.
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Solving the Problem\end{tabular}}
What happens if we extend our regular expressions
\begin{center}
\begin{tabular}{rcl}
\bl{$r$} & \bl{$::=$} & \bl{\ldots}\\
& \bl{$\mid$} & \bl{$r\{n\}$}\\
& \bl{$\mid$} & \bl{$r?$}
\end{tabular}
\end{center}
What is their meaning? What are the cases for \bl{$nullable$} and \bl{$der$}?
\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}
\mbox{}\\[-13mm]
\begin{tikzpicture}[y=.2cm, x=.12cm]
%axis
\draw (0,0) -- coordinate (x axis mid) (70,0);
\draw (0,0) -- coordinate (y axis mid) (0,30);
%ticks
\foreach \x in {0,10,...,70}
\draw (\x,1pt) -- (\x,-3pt)
node[anchor=north] {\x};
\foreach \y in {0,5,...,30}
\draw (1pt,\y) -- (-3pt,\y)
node[anchor=east] {\y};
%labels
\node[below=0.6cm] at (x axis mid) {\bl{a}s};
\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
%plots
\draw[color=blue] plot[mark=*, mark options={fill=white}]
file {re-python.data};
\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
file {re1.data};
\draw[color=green] plot[mark=square*, mark options={fill=white} ]
file {re2a.data};
\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ]
file {re-ruby.data};
%legend
\begin{scope}[shift={(4,20)}]
\draw[color=blue] (0,0) --
plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Python};
\draw[yshift=-\baselineskip, color=brown] (0,0) --
plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Ruby};
\draw[yshift=\baselineskip, color=red] (0,0) --
plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Scala V1};
\draw[yshift=2\baselineskip, color=green] (0,0) --
plot[mark=square*, mark options={fill=white}] (0.25,0) -- (0.5,0)
node[right]{\small Scala V2};
\end{scope}
\end{tikzpicture}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}
\mbox{}\\[-13mm]
\begin{tabular}{@ {\hspace{-5mm}}l}
\begin{tikzpicture}[y=.2cm, x=.01cm]
%axis
\draw (0,0) -- coordinate (x axis mid) (1000,0);
\draw (0,0) -- coordinate (y axis mid) (0,30);
%ticks
\foreach \x in {0,100,...,1000}
\draw (\x,1pt) -- (\x,-3pt)
node[anchor=north] {\x};
\foreach \y in {0,5,...,30}
\draw (1pt,\y) -- (-3pt,\y)
node[anchor=east] {\y};
%labels
\node[below=0.6cm] at (x axis mid) {\bl{a}s};
\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
%plots
\draw[color=blue] plot[mark=*, mark options={fill=white}]
file {re-python.data};
\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
file {re1.data};
\draw[color=green] plot[mark=square*, mark options={fill=white} ]
file {re2b.data};
\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ]
file {re-ruby.data};
%legend
\begin{scope}[shift={(100,20)}]
\draw[color=blue] (0,0) --
plot[mark=*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Python};
\draw[yshift=-13, color=brown] (0,0) --
plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Ruby};
\draw[yshift=13, color=red] (0,0) --
plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Scala V1};
\draw[yshift=26, color=green] (0,0) --
plot[mark=square*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Scala V2};
\end{scope}
\end{tikzpicture}
\end{tabular}
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Examples\end{tabular}}
Recall the example of \bl{$r \dn ((a \cdot b) + b)^*$} with
\begin{center}
\begin{tabular}{l}
\bl{$der\,a\,r = ((\epsilon \cdot b) + \varnothing) \cdot r$}\\
\bl{$der\,b\,r = ((\varnothing \cdot b) + \epsilon)\cdot r$}
\end{tabular}
\end{center}
What are these regular expressions equal to?
\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}
\mbox{}\\[-9mm]
\begin{tabular}{@ {\hspace{-5mm}}l}
\begin{tikzpicture}[y=.2cm, x=.0008cm]
%axis
\draw (0,0) -- coordinate (x axis mid) (12000,0);
\draw (0,0) -- coordinate (y axis mid) (0,30);
%ticks
\foreach \x in {0,2000,...,12000}
\draw (\x,1pt) -- (\x,-3pt)
node[anchor=north] {\x};
\foreach \y in {0,5,...,30}
\draw (1pt,\y) -- (-3pt,\y)
node[anchor=east] {\y};
%labels
\node[below=0.6cm] at (x axis mid) {\bl{a}s};
\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
%plots
\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
file {re1.data};
\draw[color=green] plot[mark=square*, mark options={fill=white} ]
file {re2b.data};
\draw[color=black] plot[mark=square*, mark options={fill=white} ]
file {re3.data};
%legend
\begin{scope}[shift={(2000,20)}]
\draw[color=red] (0,0) --
plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Scala V1};
\draw[yshift=13, color=green] (0,0) --
plot[mark=square*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Scala V2};
\draw[yshift=26, color=black] (0,0) --
plot[mark=square*, mark options={fill=white}] (0.25,0) -- (50,0)
node[right]{\small Scala V3};
\end{scope}
\end{tikzpicture}
\end{tabular}
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
\begin{itemize}
\item symbols, digits
\item identifiers
\item numbers (non-leading zeros)
\item keywords
\item comments
\end{itemize}\bigskip
\mbox{}\hfill\bl{\url{http://www.regexper.com}}
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}
A language (a set of strings) is \alert{regular} iff there exists
a regular expression that recognises all its strings.\bigskip\bigskip\pause
Do you think there are languages that are {\bf not} regular?
\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
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\end{document}
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