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(*<*)
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theory Slides1
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imports "~~/src/HOL/Library/LaTeXsugar"
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begin
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notation (latex output)
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set ("_") and
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Cons ("_::/_" [66,65] 65)
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(*>*)
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text_raw {*
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%\renewcommand{\slidecaption}{Cambridge, 9 November 2010}
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\renewcommand{\slidecaption}{Nijmegen, 25 August 2011}
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\Large A Formalisation of the\\[-4mm]
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\Large Myhill-Nerode Theorem based on\\[-4mm]
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\Large Regular Expressions\\[-4mm]
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\Large (Proof Pearl)\\[0mm]
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\end{tabular}}
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\begin{center}
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\begin{tabular}{c@ {\hspace{15mm}}c}
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\includegraphics[scale=0.034]{chunhan.jpg} &
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\includegraphics[scale=0.034]{xingyuan.jpg}\\[-5mm]
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\begin{center}
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\small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
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University of Science and Technology in Nanjing
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\end{center}
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\begin{center}
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\small Christian Urban\\
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TU Munich
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\frametitle{}
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\begin{textblock}{12.9}(1.5,3.2)
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\begin{block}{}
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\begin{minipage}{12.4cm}\raggedright
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\large I want to teach \alert{students}\\
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with theorem provers (induction).
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\end{minipage}
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\end{block}
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\end{textblock}\pause
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\mbox{}\\[35mm]\mbox{}
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\begin{itemize}
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\item \only<2>{\smath{\text{fib}}, \smath{\text{even}} and \smath{\text{odd}}}%
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\only<3->{\textcolor{red}{\sout{\textcolor{black}%
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{\smath{\text{fib}}, \smath{\text{even}} and \smath{\text{odd}}}}}}\medskip
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\item<3-> formal language theory \\
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\mbox{}\;\;@{text "\<Rightarrow>"} nice textbooks: Kozen, Hopcroft \& Ullman
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\end{itemize}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\begin{center}
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\huge\bf\textcolor{gray}{in Nuprl}
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\end{center}
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\begin{itemize}
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\item Constable, Jackson, Naumov, Uribe\medskip
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\item \alert{18 months} for automata theory, Hopcroft \& Ullman chapters 1--11 (including Myhill-Nerode)
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\end{itemize}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\begin{center}
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\huge\bf\textcolor{gray}{in Coq}
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\end{center}
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\begin{itemize}
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\item Filli\^atre, Briais, Braibant and others
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\item multi-year effort; a number of results in automata theory, e.g.\medskip
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\begin{itemize}
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\item Kleene's thm.~by Filli\^atre (\alert{``rather big''})
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\item automata theory by Briais (5400 loc)
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\item Braibant ATBR library, including Myhill-Nerode ($>\!\!\!>$2000 loc)
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\item Mirkin's partial derivative automaton construction (10600 loc)
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\end{itemize}
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\end{itemize}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\mbox{}\\[-10mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in HOL}
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\end{center}
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\begin{itemize}
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\item automata @{text "\<Rightarrow>"} graphs, matrices, functions
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\item<2-> combining automata/graphs
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&
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\onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
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&
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\end{center}\medskip
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\only<4-5>{
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\begin{tabular}{@ {}l@ {}}
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disjoint union:\\[2mm]
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\smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
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\end{tabular}}
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\end{itemize}
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\only<5>{
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\medskip
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\begin{minipage}{14cm}\raggedright
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Problems with definition for regularity (Slind):\bigskip\\
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\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
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\end{minipage}
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\medskip
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\only<6->{A solution:\;\;\smath{\text{nat}} \;@{text "\<Rightarrow>"}\; state nodes\medskip}
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\only<7->{You have to \alert{\uline{rename}} states!}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\mbox{}\\[-10mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in HOL}
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\end{center}
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\begin{itemize}
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\item Kozen's paper proof of Myhill-Nerode:\\
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\hspace{2cm}requires absence of \alert{inaccessible states}
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\end{itemize}\bigskip\bigskip
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\begin{center}
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\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
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\end{center}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{}
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\mbox{}\\[25mm]\mbox{}
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\begin{textblock}{13.9}(0.7,1.2)
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{\bf Definition:}\smallskip\\
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A language \smath{A} is \alert{regular}, provided there exists a\\
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regular expression that matches all strings of \smath{A}.
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\end{textblock}\pause
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{\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
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Do we lose anything?\pause
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\begin{itemize}
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\item pumping lemma\pause
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\item closure under complementation\pause
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\item \only<6>{regular expression matching}%
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\only<7->{\textcolor{red}{\sout{\textcolor{black}{regular expression matching}}}}
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\item<8-> most textbooks are about automata
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\end{itemize}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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\begin{itemize}
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\item provides necessary and suf\!ficient conditions for a language
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being regular (pumping lemma only necessary)\medskip
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\item will help with closure properties of regular languages\bigskip\pause
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\item key is the equivalence relation:\smallskip
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\begin{center}
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\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
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\end{itemize}
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\only<2->
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Two directions:\smallskip\\
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1.) \\
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2.) \\
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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\mbox{}\\[5cm]
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\begin{itemize}
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\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
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\end{itemize}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Regular Languages}
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\begin{itemize}
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\item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
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such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
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349 |
|
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|
350 |
\item Myhill-Nerode:
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351 |
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|
352 |
\begin{center}
|
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|
353 |
\begin{tabular}{l}
|
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|
354 |
finite $\Rightarrow$ regular\\
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|
355 |
\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
|
|
|
356 |
regular $\Rightarrow$ finite\\
|
|
|
357 |
\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
|
358 |
\end{tabular}
|
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|
359 |
\end{center}
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|
360 |
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|
361 |
\end{itemize}
|
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362 |
|
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|
363 |
\end{frame}}
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364 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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365 |
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|
366 |
*}
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367 |
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368 |
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369 |
text_raw {*
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370 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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371 |
\mode<presentation>{
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|
372 |
\begin{frame}[c]
|
|
|
373 |
\frametitle{\LARGE Final States}
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|
374 |
|
|
|
375 |
\mbox{}\\[3cm]
|
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|
376 |
|
|
|
377 |
\begin{itemize}
|
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|
378 |
\item \smath{\text{final}_L\,X \dn}\\
|
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|
379 |
\smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
|
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|
380 |
\smallskip
|
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|
381 |
\item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
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382 |
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|
383 |
\end{itemize}
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384 |
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385 |
\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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387 |
*}
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388 |
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389 |
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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392 |
\mode<presentation>{
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393 |
\begin{frame}[c]
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|
394 |
\frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
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395 |
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396 |
\smath{L = \{[c]\}}
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397 |
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398 |
\begin{tabular}{@ {\hspace{-7mm}}cc}
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399 |
\begin{tabular}{c}
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400 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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401 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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402 |
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403 |
%\draw[help lines] (0,0) grid (3,2);
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404 |
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|
405 |
\node[state,initial] (q_0) {$R_1$};
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|
406 |
\node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
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|
407 |
\node[state] (q_2) [below right of=q_0] {$R_3$};
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|
408 |
|
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|
409 |
\path[->] (q_0) edge node {c} (q_1)
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410 |
edge node [swap] {$\Sigma-{c}$} (q_2)
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411 |
(q_2) edge [loop below] node {$\Sigma$} ()
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412 |
(q_1) edge node {$\Sigma$} (q_2);
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413 |
\end{tikzpicture}
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414 |
\end{tabular}
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415 |
&
|
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|
416 |
\begin{tabular}[t]{ll}
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417 |
\\[-20mm]
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|
418 |
\multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
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419 |
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|
420 |
\smath{R_1}: & \smath{\{[]\}}\\
|
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|
421 |
\smath{R_2}: & \smath{\{[c]\}}\\
|
|
|
422 |
\smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
|
|
|
423 |
\multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}
|
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424 |
\end{tabular}
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|
425 |
|
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|
426 |
\end{tabular}
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427 |
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|
428 |
\end{frame}}
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429 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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430 |
*}
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431 |
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432 |
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433 |
text_raw {*
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434 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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435 |
\mode<presentation>{
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|
436 |
\begin{frame}[c]
|
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|
437 |
\frametitle{\LARGE Systems of Equations}
|
|
|
438 |
|
|
|
439 |
Inspired by a method of Brzozowski\;'64, we can build an equational system
|
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|
440 |
characterising the equivalence classes:
|
|
|
441 |
|
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|
442 |
\begin{center}
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|
443 |
\begin{tabular}{@ {\hspace{-20mm}}c}
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|
444 |
\\[-13mm]
|
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445 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
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|
446 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
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|
447 |
|
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|
448 |
%\draw[help lines] (0,0) grid (3,2);
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449 |
|
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|
450 |
\node[state,initial] (p_0) {$R_1$};
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|
451 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
452 |
|
|
|
453 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
454 |
edge [loop above] node {b} ()
|
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|
455 |
(p_1) edge [loop above] node {a} ()
|
|
|
456 |
edge [bend left] node {b} (p_0);
|
|
|
457 |
\end{tikzpicture}\\
|
|
|
458 |
\\[-13mm]
|
|
|
459 |
\end{tabular}
|
|
|
460 |
\end{center}
|
|
|
461 |
|
|
|
462 |
\begin{center}
|
|
|
463 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
|
464 |
& \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
|
|
|
465 |
& \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
|
|
|
466 |
\onslide<3->{we can prove}
|
|
|
467 |
& \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
|
|
|
468 |
& \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
|
|
|
469 |
& \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
|
|
|
470 |
& \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
|
|
|
471 |
\end{tabular}
|
|
|
472 |
\end{center}
|
|
|
473 |
|
|
|
474 |
\end{frame}}
|
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|
475 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
476 |
*}
|
|
|
477 |
|
|
|
478 |
text_raw {*
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|
479 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
480 |
\mode<presentation>{
|
|
|
481 |
\begin{frame}<1>[t]
|
|
|
482 |
\small
|
|
|
483 |
|
|
|
484 |
\begin{center}
|
|
|
485 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
486 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
487 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
488 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
489 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
490 |
|
|
|
491 |
& & & \onslide<2->{by Arden}\\
|
|
|
492 |
|
|
|
493 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
|
494 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
495 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
496 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
497 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
498 |
|
|
|
499 |
& & & \onslide<4->{by Arden}\\
|
|
|
500 |
|
|
|
501 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
502 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
503 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
504 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
505 |
|
|
|
506 |
& & & \onslide<5->{by substitution}\\
|
|
|
507 |
|
|
|
508 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
|
509 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
510 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
511 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
512 |
|
|
|
513 |
& & & \onslide<6->{by Arden}\\
|
|
|
514 |
|
|
|
515 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
516 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
517 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
518 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
519 |
|
|
|
520 |
& & & \onslide<7->{by substitution}\\
|
|
|
521 |
|
|
|
522 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
523 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
524 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
525 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
526 |
\cdot a\cdot a^\star}}\\
|
|
|
527 |
\end{tabular}
|
|
|
528 |
\end{center}
|
|
|
529 |
|
|
|
530 |
\end{frame}}
|
|
|
531 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
532 |
*}
|
|
|
533 |
|
|
|
534 |
text_raw {*
|
|
|
535 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
536 |
\mode<presentation>{
|
|
|
537 |
\begin{frame}[c]
|
|
|
538 |
\frametitle{\LARGE A Variant of Arden's Lemma}
|
|
|
539 |
|
|
|
540 |
{\bf Arden's Lemma:}\smallskip
|
|
|
541 |
|
|
|
542 |
If \smath{[] \not\in A} then
|
|
|
543 |
\begin{center}
|
|
|
544 |
\smath{X = X; A + \text{something}}
|
|
|
545 |
\end{center}
|
|
|
546 |
has the (unique) solution
|
|
|
547 |
\begin{center}
|
|
|
548 |
\smath{X = \text{something} ; A^\star}
|
|
|
549 |
\end{center}
|
|
|
550 |
|
|
|
551 |
|
|
|
552 |
\end{frame}}
|
|
|
553 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
554 |
*}
|
|
|
555 |
|
|
|
556 |
|
|
|
557 |
text_raw {*
|
|
|
558 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
559 |
\mode<presentation>{
|
|
|
560 |
\begin{frame}<1->[t]
|
|
|
561 |
\small
|
|
|
562 |
|
|
|
563 |
\begin{center}
|
|
|
564 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
565 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
566 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
567 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
568 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
569 |
|
|
|
570 |
& & & \onslide<2->{by Arden}\\
|
|
|
571 |
|
|
|
572 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
|
573 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
574 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
575 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
576 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
577 |
|
|
|
578 |
& & & \onslide<4->{by Arden}\\
|
|
|
579 |
|
|
|
580 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
581 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
582 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
583 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
584 |
|
|
|
585 |
& & & \onslide<5->{by substitution}\\
|
|
|
586 |
|
|
|
587 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
|
588 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
589 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
590 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
591 |
|
|
|
592 |
& & & \onslide<6->{by Arden}\\
|
|
|
593 |
|
|
|
594 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
595 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
596 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
597 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
598 |
|
|
|
599 |
& & & \onslide<7->{by substitution}\\
|
|
|
600 |
|
|
|
601 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
602 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
603 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
604 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
605 |
\cdot a\cdot a^\star}}\\
|
|
|
606 |
\end{tabular}
|
|
|
607 |
\end{center}
|
|
|
608 |
|
|
|
609 |
\only<8->{
|
|
|
610 |
\begin{textblock}{6}(2.5,4)
|
|
|
611 |
\begin{block}{}
|
|
|
612 |
\begin{minipage}{8cm}\raggedright
|
|
|
613 |
|
|
|
614 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
|
|
615 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
616 |
|
|
|
617 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
618 |
|
|
|
619 |
\node[state,initial] (p_0) {$R_1$};
|
|
|
620 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
621 |
|
|
|
622 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
623 |
edge [loop above] node {b} ()
|
|
|
624 |
(p_1) edge [loop above] node {a} ()
|
|
|
625 |
edge [bend left] node {b} (p_0);
|
|
|
626 |
\end{tikzpicture}
|
|
|
627 |
|
|
|
628 |
\end{minipage}
|
|
|
629 |
\end{block}
|
|
|
630 |
\end{textblock}}
|
|
|
631 |
|
|
|
632 |
\end{frame}}
|
|
|
633 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
634 |
*}
|
|
|
635 |
|
|
|
636 |
text_raw {*
|
|
|
637 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
638 |
\mode<presentation>{
|
|
|
639 |
\begin{frame}[c]
|
|
|
640 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
|
|
641 |
|
|
|
642 |
\begin{itemize}
|
|
|
643 |
\item The algorithm must terminate: Arden makes one equation smaller;
|
|
|
644 |
substitution deletes one variable from the right-hand sides.\bigskip
|
|
|
645 |
|
|
|
646 |
\item We need to maintain the invariant that Arden is applicable
|
|
|
647 |
(if \smath{[] \not\in A} then \ldots):\medskip
|
|
|
648 |
|
|
|
649 |
\begin{center}\small
|
|
|
650 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
651 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
|
652 |
\smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
|
|
|
653 |
|
|
|
654 |
& & & by Arden\\
|
|
|
655 |
|
|
|
656 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
|
657 |
\smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
|
|
|
658 |
\end{tabular}
|
|
|
659 |
\end{center}
|
|
|
660 |
|
|
|
661 |
\end{itemize}
|
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|
662 |
|
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|
663 |
|
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|
664 |
\end{frame}}
|
|
|
665 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
666 |
*}
|
|
|
667 |
|
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|
668 |
|
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|
669 |
text_raw {*
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|
670 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
671 |
\mode<presentation>{
|
|
|
672 |
\begin{frame}[c]
|
|
|
673 |
\frametitle{\LARGE Other Direction}
|
|
|
674 |
|
|
|
675 |
One has to prove
|
|
|
676 |
|
|
|
677 |
\begin{center}
|
|
|
678 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
|
679 |
\end{center}
|
|
|
680 |
|
|
|
681 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
682 |
of thinking (by Chunhan), one can prove that if
|
|
|
683 |
|
|
|
684 |
\begin{center}
|
|
|
685 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
|
|
|
686 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
|
|
|
687 |
\end{center}
|
|
|
688 |
|
|
|
689 |
then
|
|
|
690 |
|
|
|
691 |
\begin{center}
|
|
|
692 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
|
|
|
693 |
\end{center}
|
|
|
694 |
|
|
|
695 |
|
|
|
696 |
|
|
|
697 |
\end{frame}}
|
|
|
698 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
699 |
*}
|
|
|
700 |
|
|
|
701 |
text_raw {*
|
|
|
702 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
703 |
\mode<presentation>{
|
|
|
704 |
\begin{frame}[c]
|
|
|
705 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
706 |
|
|
|
707 |
\begin{itemize}
|
|
|
708 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
|
|
709 |
\bigskip\pause
|
|
|
710 |
\item regular languages are closed under complementation; this is easy
|
|
|
711 |
\begin{center}
|
|
|
712 |
\smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
|
|
|
713 |
\end{center}\pause\bigskip
|
|
|
714 |
|
|
|
715 |
\item if you want to do regular expression matching (see Scott's paper)\pause\bigskip
|
|
|
716 |
|
|
|
717 |
\item I cannot yet give definite numbers
|
|
|
718 |
\end{itemize}
|
|
|
719 |
|
|
|
720 |
\only<2>{
|
|
|
721 |
\begin{textblock}{10}(4,14)
|
|
|
722 |
\small
|
|
|
723 |
\smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
|
|
|
724 |
\end{textblock}
|
|
|
725 |
}
|
|
|
726 |
|
|
|
727 |
|
|
|
728 |
|
|
|
729 |
\end{frame}}
|
|
|
730 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
731 |
*}
|
|
|
732 |
|
|
|
733 |
|
|
|
734 |
|
|
|
735 |
text_raw {*
|
|
|
736 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
737 |
\mode<presentation>{
|
|
|
738 |
\begin{frame}[c]
|
|
|
739 |
\frametitle{\LARGE What We Have Not Achieved}
|
|
|
740 |
|
|
|
741 |
\begin{itemize}
|
|
|
742 |
\item regular expressions are not good if you look for a minimal
|
|
|
743 |
one for a language (DFAs have this notion)\pause\bigskip
|
|
|
744 |
|
|
|
745 |
\item Is there anything to be said about context free languages:\medskip
|
|
|
746 |
|
|
|
747 |
\begin{quote}
|
|
|
748 |
A context free language is where every string can be recognised by
|
|
|
749 |
a pushdown automaton.
|
|
|
750 |
\end{quote}
|
|
|
751 |
\end{itemize}
|
|
|
752 |
|
|
|
753 |
\end{frame}}
|
|
|
754 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
755 |
*}
|
|
|
756 |
|
|
|
757 |
|
|
|
758 |
text_raw {*
|
|
|
759 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
760 |
\mode<presentation>{
|
|
|
761 |
\begin{frame}[c]
|
|
|
762 |
\frametitle{\LARGE Conclusion}
|
|
|
763 |
|
|
|
764 |
\begin{itemize}
|
|
|
765 |
\item on balance regular expression are superior
|
|
|
766 |
to DFAs, in my opinion\bigskip
|
|
|
767 |
|
|
|
768 |
\item I cannot think of a reason to not teach regular languages
|
|
|
769 |
to students this way (!?)\bigskip
|
|
|
770 |
|
|
|
771 |
\item I have never ever seen a proof of Myhill-Nerode based on
|
|
|
772 |
regular expressions\bigskip
|
|
|
773 |
|
|
|
774 |
\item no application, but lots of fun\bigskip
|
|
|
775 |
|
|
|
776 |
\item great source of examples
|
|
|
777 |
|
|
|
778 |
\end{itemize}
|
|
|
779 |
|
|
|
780 |
\end{frame}}
|
|
|
781 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
782 |
*}
|
|
|
783 |
|
|
|
784 |
(*<*)
|
|
|
785 |
end
|
|
|
786 |
(*>*) |