--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides1.thy Tue Aug 23 00:24:10 2011 +0000
@@ -0,0 +1,893 @@
+(*<*)
+theory Slides1
+imports "~~/src/HOL/Library/LaTeXsugar"
+begin
+
+notation (latex output)
+ set ("_") and
+ Cons ("_::/_" [66,65] 65)
+
+(*>*)
+
+
+text_raw {*
+ %\renewcommand{\slidecaption}{Cambridge, 9 November 2010}
+ \renewcommand{\slidecaption}{Nijmegen, 25 August 2011}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}
+ \frametitle{%
+ \begin{tabular}{@ {}c@ {}}
+ \Large A Formalisation of the\\[-4mm]
+ \Large Myhill-Nerode Theorem based on\\[-4mm]
+ \Large Regular Expressions\\[-4mm]
+ \Large (Proof Pearl)\\[0mm]
+ \end{tabular}}
+
+ \begin{center}
+ \begin{tabular}{c@ {\hspace{15mm}}c}
+ \includegraphics[scale=0.034]{chunhan.jpg} &
+ \includegraphics[scale=0.034]{xingyuan.jpg}\\[-5mm]
+ \end{tabular}
+ \end{center}
+
+
+ \begin{center}
+ \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
+ University of Science and Technology in Nanjing
+ \end{center}
+
+ \begin{center}
+ \small Christian Urban\\
+ TU Munich
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{In Most Textbooks\ldots}
+
+ \begin{itemize}
+ \item A \alert{regular language} is one where there is a DFA that
+ recognises it.\bigskip\pause
+ \end{itemize}
+
+
+ I can think of three reasons why this is a good definition:\medskip
+ \begin{itemize}
+ \item string matching via DFAs (yacc)
+ \item pumping lemma
+ \item closure properties of regular languages (closed under complement)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
+
+ \begin{center}
+ \huge\bf\textcolor{gray}{in Nuprl}
+ \end{center}
+
+ \begin{itemize}
+ \item Constable, Jackson, Naumov, Uribe\medskip
+ \item \alert{18 months} for automata theory, Hopcroft \& Ullman chapters 1--11 (including Myhill-Nerode)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
+
+ \begin{center}
+ \huge\bf\textcolor{gray}{in Coq}
+ \end{center}
+
+ \begin{itemize}
+ \item Filli\^atre, Briais, Braibant and others
+ \item multi-year effort; a number of results in automata theory, e.g.\medskip
+ \begin{itemize}
+ \item Kleene's thm.~by Filli\^atre (\alert{``rather big''})
+ \item automata theory by Briais (5400 loc)
+ \item Braibant ATBR library, including Myhill-Nerode ($>\!\!\!>$2000 loc)
+ \item Mirkin's partial derivative automaton construction (10600 loc)
+ \end{itemize}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
+ \mbox{}\\[-10mm]\mbox{}
+
+ \begin{center}
+ \huge\bf\textcolor{gray}{in HOL}
+ \end{center}
+
+ \begin{itemize}
+ \item automata @{text "\<Rightarrow>"} graphs, matrices, functions
+ \item<2-> combining automata/graphs
+
+ \onslide<2->{
+ \begin{center}
+ \begin{tabular}{ccc}
+ \begin{tikzpicture}[scale=1]
+ %\draw[step=2mm] (-1,-1) grid (1,1);
+
+ \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
+ \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
+
+ \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \draw (-0.6,0.0) node {\small$A_1$};
+ \draw ( 0.6,0.0) node {\small$A_2$};
+ \end{tikzpicture}}
+
+ &
+
+ \onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
+
+ &
+
+ \onslide<3->{\begin{tikzpicture}[scale=1]
+ %\draw[step=2mm] (-1,-1) grid (1,1);
+
+ \draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
+ \draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
+
+ \node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (B) at ( 0.2,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+ \node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
+
+ \draw (C) to [red, very thick, bend left=45] (B);
+ \draw (D) to [red, very thick, bend right=45] (B);
+
+ \draw (-0.6,0.0) node {\small$A_1$};
+ \draw ( 0.6,0.0) node {\small$A_2$};
+ \end{tikzpicture}}
+
+ \end{tabular}
+ \end{center}\medskip
+
+ \only<4-5>{
+ \begin{tabular}{@ {}l@ {}}
+ disjoint union:\\[2mm]
+ \smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
+ \end{tabular}}
+ \end{itemize}
+
+ \only<5>{
+ \begin{textblock}{13.9}(0.7,7.7)
+ \begin{block}{}
+ \medskip
+ \begin{minipage}{14cm}\raggedright
+ Problems with definition for regularity (Slind):\bigskip\\
+ \smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
+ \end{minipage}
+ \end{block}
+ \end{textblock}}
+ \medskip
+
+ \only<6->{A solution:\;\;\smath{\text{nat}} @{text "\<Rightarrow>"} state nodes}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
+ \mbox{}\\[-10mm]\mbox{}
+
+ \begin{center}
+ \huge\bf\textcolor{gray}{in HOL}
+ \end{center}
+
+ \begin{itemize}
+ \item Kozen's proof of Myhill-Nerode:\\
+ \hspace{5cm}\alert{inaccessible states}
+ \end{itemize}\bigskip\bigskip
+
+ \begin{center}
+ \smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Regular Expressions}
+ \mbox{}\\[20mm]\mbox{}
+
+ \begin{textblock}{13.9}(0.7,2.2)
+ \begin{block}{}
+ \begin{minipage}{13.4cm}\raggedright
+ {\bf Definition:}\smallskip\\
+
+ A language \smath{A} is \alert{regular}, provided there exists a\\
+ regular expression that matches all strings of \smath{A}.
+ \end{minipage}
+ \end{block}
+ \end{textblock}\pause
+
+ {\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
+
+ What we might lose?\pause
+ \begin{itemize}\renewcommand{\ULthickness}{2pt}
+ \item pumping lemma\pause
+ \item closure under complementation\pause
+ \item \only<6>{regular expression matching}%
+ \only<7>{\textcolor{red}{\sout{\textcolor{black}{regular expression matching}}}}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Regular Expressions}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Regular Expression Matching}
+
+ \begin{itemize}
+ \item Harper in JFP'99: ``Functional Pearl: Proof- Directed Debugging''\medskip
+ \item Yi in JFP'06: ``Educational Pearl: `Proof-Directed Debugging' revisited
+ for a first-order version''\medskip
+ \item Owens et al in JFP'09: ``Regular-expression derivatives re-examined''\bigskip\pause
+
+ \begin{quote}\small
+ ``Unfortunately, regular expression derivatives have been lost in the
+ sands of time, and few computer scientists are aware of them.''
+ \end{quote}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \begin{itemize}
+ \item provides necessary and suf\!ficient conditions for a language
+ being regular (pumping lemma only necessary)\medskip
+
+ \item will help with closure properties of regular languages\bigskip\pause
+
+ \item key is the equivalence relation:\smallskip
+ \begin{center}
+ \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
+ \end{center}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \mbox{}\\[5cm]
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Equivalence Classes}
+
+ \begin{itemize}
+ \item \smath{L = []}
+ \begin{center}
+ \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
+ \end{center}\bigskip\bigskip
+
+ \item \smath{L = [c]}
+ \begin{center}
+ \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
+ \end{center}\bigskip\bigskip
+
+ \item \smath{L = \varnothing}
+ \begin{center}
+ \smath{\Big\{U\!N\!IV\Big\}}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Regular Languages}
+
+ \begin{itemize}
+ \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
+ such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
+
+ \item Myhill-Nerode:
+
+ \begin{center}
+ \begin{tabular}{l}
+ finite $\Rightarrow$ regular\\
+ \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
+ regular $\Rightarrow$ finite\\
+ \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Final States}
+
+ \mbox{}\\[3cm]
+
+ \begin{itemize}
+ \item \smath{\text{final}_L\,X \dn}\\
+ \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
+ \smallskip
+ \item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
+
+ \smath{L = \{[c]\}}
+
+ \begin{tabular}{@ {\hspace{-7mm}}cc}
+ \begin{tabular}{c}
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (q_0) {$R_1$};
+ \node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
+ \node[state] (q_2) [below right of=q_0] {$R_3$};
+
+ \path[->] (q_0) edge node {c} (q_1)
+ edge node [swap] {$\Sigma-{c}$} (q_2)
+ (q_2) edge [loop below] node {$\Sigma$} ()
+ (q_1) edge node {$\Sigma$} (q_2);
+ \end{tikzpicture}
+ \end{tabular}
+ &
+ \begin{tabular}[t]{ll}
+ \\[-20mm]
+ \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
+
+ \smath{R_1}: & \smath{\{[]\}}\\
+ \smath{R_2}: & \smath{\{[c]\}}\\
+ \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
+ \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}
+ \end{tabular}
+
+ \end{tabular}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Systems of Equations}
+
+ Inspired by a method of Brzozowski\;'64, we can build an equational system
+ characterising the equivalence classes:
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-20mm}}c}
+ \\[-13mm]
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (p_0) {$R_1$};
+ \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
+
+ \path[->] (p_0) edge [bend left] node {a} (p_1)
+ edge [loop above] node {b} ()
+ (p_1) edge [loop above] node {a} ()
+ edge [bend left] node {b} (p_0);
+ \end{tikzpicture}\\
+ \\[-13mm]
+ \end{tabular}
+ \end{center}
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+ & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
+ & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
+ \onslide<3->{we can prove}
+ & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
+ & \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
+ & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
+ & \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
+ \small
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+ & & & \onslide<2->{by Arden}\\
+
+ \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
+ & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
+ & \only<2>{\smath{R_1; a + R_2; a}}%
+ \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<4->{by Arden}\\
+
+ \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<5->{by substitution}\\
+
+ \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<6->{by Arden}\\
+
+ \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<7->{by substitution}\\
+
+ \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
+ \cdot a\cdot a^\star}}\\
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE A Variant of Arden's Lemma}
+
+ {\bf Arden's Lemma:}\smallskip
+
+ If \smath{[] \not\in A} then
+ \begin{center}
+ \smath{X = X; A + \text{something}}
+ \end{center}
+ has the (unique) solution
+ \begin{center}
+ \smath{X = \text{something} ; A^\star}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \small
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+ & & & \onslide<2->{by Arden}\\
+
+ \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
+ & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
+ & \only<2>{\smath{R_1; a + R_2; a}}%
+ \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<4->{by Arden}\\
+
+ \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<5->{by substitution}\\
+
+ \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<6->{by Arden}\\
+
+ \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<7->{by substitution}\\
+
+ \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
+ \cdot a\cdot a^\star}}\\
+ \end{tabular}
+ \end{center}
+
+ \only<8->{
+ \begin{textblock}{6}(2.5,4)
+ \begin{block}{}
+ \begin{minipage}{8cm}\raggedright
+
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (p_0) {$R_1$};
+ \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
+
+ \path[->] (p_0) edge [bend left] node {a} (p_1)
+ edge [loop above] node {b} ()
+ (p_1) edge [loop above] node {a} ()
+ edge [bend left] node {b} (p_0);
+ \end{tikzpicture}
+
+ \end{minipage}
+ \end{block}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Equ's Solving Algorithm}
+
+ \begin{itemize}
+ \item The algorithm must terminate: Arden makes one equation smaller;
+ substitution deletes one variable from the right-hand sides.\bigskip
+
+ \item We need to maintain the invariant that Arden is applicable
+ (if \smath{[] \not\in A} then \ldots):\medskip
+
+ \begin{center}\small
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+ \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
+
+ & & & by Arden\\
+
+ \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+ \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Equ's Solving Algorithm}
+
+ \begin{itemize}
+ \item The algorithm is still a bit hairy to formalise because of our set-representation
+ for equations:
+
+ \begin{center}
+ \begin{tabular}{ll}
+ \smath{\big\{ (X, \{(Y_1, r_1), (Y_2, r_2), \ldots\}),}\\
+ \mbox{}\hspace{5mm}\smath{\ldots}\\
+ & \smath{\big\}}
+ \end{tabular}
+ \end{center}\bigskip\pause
+
+ \small
+ they are generated from \smath{U\!N\!IV /\!/ \approx_L}
+
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Other Direction}
+
+ One has to prove
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+ \end{center}
+
+ by induction on \smath{r}. Not trivial, but after a bit
+ of thinking (by Chunhan), one can prove that if
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
+ \end{center}
+
+ then
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
+ \end{center}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE What Have We Achieved?}
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+ \bigskip\pause
+ \item regular languages are closed under complementation; this is easy
+ \begin{center}
+ \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
+ \end{center}\pause\bigskip
+
+ \item if you want to do regular expression matching (see Scott's paper)\pause\bigskip
+
+ \item I cannot yet give definite numbers
+ \end{itemize}
+
+ \only<2>{
+ \begin{textblock}{10}(4,14)
+ \small
+ \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
+ \end{textblock}
+ }
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Examples}
+
+ \begin{itemize}
+ \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
+ \begin{quote}\small
+ \begin{tabular}{lcl}
+ \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
+ \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
+ \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
+ \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
+ \end{tabular}
+ \end{quote}
+
+ \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
+ \begin{quote}\small
+ \begin{tabular}{lcl}
+ \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
+ \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
+ \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
+ \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
+ & \smath{\vdots} &\\
+ \end{tabular}
+ \end{quote}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE What We Have Not Achieved}
+
+ \begin{itemize}
+ \item regular expressions are not good if you look for a minimal
+ one for a language (DFAs have this notion)\pause\bigskip
+
+ \item Is there anything to be said about context free languages:\medskip
+
+ \begin{quote}
+ A context free language is where every string can be recognised by
+ a pushdown automaton.
+ \end{quote}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Conclusion}
+
+ \begin{itemize}
+ \item on balance regular expression are superior
+ to DFAs, in my opinion\bigskip
+
+ \item I cannot think of a reason to not teach regular languages
+ to students this way (!?)\bigskip
+
+ \item I have never ever seen a proof of Myhill-Nerode based on
+ regular expressions\bigskip
+
+ \item no application, but lots of fun\bigskip
+
+ \item great source of examples
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+(*<*)
+end
+(*>*)
\ No newline at end of file
