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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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\mode<presentation>{
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\begin{frame}
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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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\end{tabular}
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\end{center}
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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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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{In Most Textbooks\ldots}
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\begin{itemize}
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\item A \alert{regular language} is one where there is a DFA that
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recognises it.\bigskip\pause
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\end{itemize}
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I can think of three reasons why this is a good definition:\medskip
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\begin{itemize}
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\item string matching via DFAs (yacc)
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\item pumping lemma
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\item closure properties of regular languages (closed under complement)
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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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text_raw {*
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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 ($>\!\!\!>$2000 loc)
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\item Mirkin's partial derivative automaton construction (10600 loc)
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\end{itemize}
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\end{itemize}
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\end{frame}}
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\mbox{}\\[-10mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in HOL}
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\end{center}
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\begin{itemize}
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\item automata @{text "\<Rightarrow>"} graphs, matrices, functions
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\item<2-> combining automata/graphs
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\onslide<2->{
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\begin{center}
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%\draw[step=2mm] (-1,-1) grid (1,1);
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\draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
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\draw (-0.6,0.0) node {\small$A_1$};
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\end{tikzpicture}}
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&
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\onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
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&
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\onslide<3->{\begin{tikzpicture}[scale=1]
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%\draw[step=2mm] (-1,-1) grid (1,1);
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\node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\draw (C) to [red, very thick, bend left=45] (B);
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\end{tikzpicture}}
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\end{tabular}
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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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\end{textblock}}
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\medskip
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\only<6->{A solution:\;\;\smath{\text{nat}} @{text "\<Rightarrow>"} state nodes}
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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 proof of Myhill-Nerode:\\
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\hspace{5cm}\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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*}
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{Regular Expressions}
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\mbox{}\\[20mm]\mbox{}
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\begin{textblock}{13.9}(0.7,2.2)
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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{minipage}
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\end{block}
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\end{textblock}\pause
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{\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
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What we might lose?\pause
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\begin{itemize}\renewcommand{\ULthickness}{2pt}
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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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\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{Regular Expressions}
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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 Expression Matching}
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\begin{itemize}
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\item Harper in JFP'99: ``Functional Pearl: Proof- Directed Debugging''\medskip
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\item Yi in JFP'06: ``Educational Pearl: `Proof-Directed Debugging' revisited
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for a first-order version''\medskip
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\item Owens et al in JFP'09: ``Regular-expression derivatives re-examined''\bigskip\pause
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\begin{quote}\small
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``Unfortunately, regular expression derivatives have been lost in the
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sands of time, and few computer scientists are aware of them.''
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\end{quote}
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\end{itemize}
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*}
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\mode<presentation>{
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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_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
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\end{center}
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\end{itemize}
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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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\begin{frame}[c]
|
|
|
360 |
\frametitle{\LARGE Equivalence Classes}
|
|
|
361 |
|
|
|
362 |
\begin{itemize}
|
|
|
363 |
\item \smath{L = []}
|
|
|
364 |
\begin{center}
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|
365 |
\smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
|
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|
366 |
\end{center}\bigskip\bigskip
|
|
|
367 |
|
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|
368 |
\item \smath{L = [c]}
|
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|
369 |
\begin{center}
|
|
|
370 |
\smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
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|
371 |
\end{center}\bigskip\bigskip
|
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|
372 |
|
|
|
373 |
\item \smath{L = \varnothing}
|
|
|
374 |
\begin{center}
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|
375 |
\smath{\Big\{U\!N\!IV\Big\}}
|
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|
376 |
\end{center}
|
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|
377 |
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|
378 |
\end{itemize}
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379 |
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|
380 |
\end{frame}}
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381 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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382 |
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383 |
*}
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384 |
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385 |
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386 |
text_raw {*
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387 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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388 |
\mode<presentation>{
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|
389 |
\begin{frame}[c]
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390 |
\frametitle{\LARGE Regular Languages}
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391 |
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|
392 |
\begin{itemize}
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|
393 |
\item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
|
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|
394 |
such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
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395 |
|
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|
396 |
\item Myhill-Nerode:
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397 |
|
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|
398 |
\begin{center}
|
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|
399 |
\begin{tabular}{l}
|
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|
400 |
finite $\Rightarrow$ regular\\
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|
401 |
\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
|
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|
402 |
regular $\Rightarrow$ finite\\
|
|
|
403 |
\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
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|
404 |
\end{tabular}
|
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|
405 |
\end{center}
|
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406 |
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407 |
\end{itemize}
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408 |
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|
409 |
\end{frame}}
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|
410 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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411 |
|
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412 |
*}
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413 |
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414 |
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415 |
text_raw {*
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416 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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417 |
\mode<presentation>{
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418 |
\begin{frame}[c]
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|
419 |
\frametitle{\LARGE Final States}
|
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420 |
|
|
|
421 |
\mbox{}\\[3cm]
|
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422 |
|
|
|
423 |
\begin{itemize}
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|
424 |
\item \smath{\text{final}_L\,X \dn}\\
|
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|
425 |
\smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
|
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|
426 |
\smallskip
|
|
|
427 |
\item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
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|
428 |
|
|
|
429 |
\end{itemize}
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|
430 |
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431 |
\end{frame}}
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432 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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433 |
*}
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434 |
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435 |
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436 |
text_raw {*
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437 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
438 |
\mode<presentation>{
|
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|
439 |
\begin{frame}[c]
|
|
|
440 |
\frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
|
|
|
441 |
|
|
|
442 |
\smath{L = \{[c]\}}
|
|
|
443 |
|
|
|
444 |
\begin{tabular}{@ {\hspace{-7mm}}cc}
|
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|
445 |
\begin{tabular}{c}
|
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|
446 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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|
447 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
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448 |
|
|
|
449 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
450 |
|
|
|
451 |
\node[state,initial] (q_0) {$R_1$};
|
|
|
452 |
\node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
|
|
|
453 |
\node[state] (q_2) [below right of=q_0] {$R_3$};
|
|
|
454 |
|
|
|
455 |
\path[->] (q_0) edge node {c} (q_1)
|
|
|
456 |
edge node [swap] {$\Sigma-{c}$} (q_2)
|
|
|
457 |
(q_2) edge [loop below] node {$\Sigma$} ()
|
|
|
458 |
(q_1) edge node {$\Sigma$} (q_2);
|
|
|
459 |
\end{tikzpicture}
|
|
|
460 |
\end{tabular}
|
|
|
461 |
&
|
|
|
462 |
\begin{tabular}[t]{ll}
|
|
|
463 |
\\[-20mm]
|
|
|
464 |
\multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
|
|
|
465 |
|
|
|
466 |
\smath{R_1}: & \smath{\{[]\}}\\
|
|
|
467 |
\smath{R_2}: & \smath{\{[c]\}}\\
|
|
|
468 |
\smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
|
|
|
469 |
\multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}
|
|
|
470 |
\end{tabular}
|
|
|
471 |
|
|
|
472 |
\end{tabular}
|
|
|
473 |
|
|
|
474 |
\end{frame}}
|
|
|
475 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
476 |
*}
|
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|
477 |
|
|
|
478 |
|
|
|
479 |
text_raw {*
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|
|
480 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
481 |
\mode<presentation>{
|
|
|
482 |
\begin{frame}[c]
|
|
|
483 |
\frametitle{\LARGE Systems of Equations}
|
|
|
484 |
|
|
|
485 |
Inspired by a method of Brzozowski\;'64, we can build an equational system
|
|
|
486 |
characterising the equivalence classes:
|
|
|
487 |
|
|
|
488 |
\begin{center}
|
|
|
489 |
\begin{tabular}{@ {\hspace{-20mm}}c}
|
|
|
490 |
\\[-13mm]
|
|
|
491 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
|
492 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
493 |
|
|
|
494 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
495 |
|
|
|
496 |
\node[state,initial] (p_0) {$R_1$};
|
|
|
497 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
498 |
|
|
|
499 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
500 |
edge [loop above] node {b} ()
|
|
|
501 |
(p_1) edge [loop above] node {a} ()
|
|
|
502 |
edge [bend left] node {b} (p_0);
|
|
|
503 |
\end{tikzpicture}\\
|
|
|
504 |
\\[-13mm]
|
|
|
505 |
\end{tabular}
|
|
|
506 |
\end{center}
|
|
|
507 |
|
|
|
508 |
\begin{center}
|
|
|
509 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
|
510 |
& \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
|
|
|
511 |
& \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
|
|
|
512 |
\onslide<3->{we can prove}
|
|
|
513 |
& \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
|
|
|
514 |
& \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
|
|
|
515 |
& \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
|
|
|
516 |
& \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
|
|
|
517 |
\end{tabular}
|
|
|
518 |
\end{center}
|
|
|
519 |
|
|
|
520 |
\end{frame}}
|
|
|
521 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
522 |
*}
|
|
|
523 |
|
|
|
524 |
text_raw {*
|
|
|
525 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
526 |
\mode<presentation>{
|
|
|
527 |
\begin{frame}<1>[t]
|
|
|
528 |
\small
|
|
|
529 |
|
|
|
530 |
\begin{center}
|
|
|
531 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
532 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
533 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
534 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
535 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
536 |
|
|
|
537 |
& & & \onslide<2->{by Arden}\\
|
|
|
538 |
|
|
|
539 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
|
540 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
541 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
542 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
543 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
544 |
|
|
|
545 |
& & & \onslide<4->{by Arden}\\
|
|
|
546 |
|
|
|
547 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
548 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
549 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
550 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
551 |
|
|
|
552 |
& & & \onslide<5->{by substitution}\\
|
|
|
553 |
|
|
|
554 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
|
555 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
556 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
557 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
558 |
|
|
|
559 |
& & & \onslide<6->{by Arden}\\
|
|
|
560 |
|
|
|
561 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
562 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
563 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
564 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
565 |
|
|
|
566 |
& & & \onslide<7->{by substitution}\\
|
|
|
567 |
|
|
|
568 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
569 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
570 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
571 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
572 |
\cdot a\cdot a^\star}}\\
|
|
|
573 |
\end{tabular}
|
|
|
574 |
\end{center}
|
|
|
575 |
|
|
|
576 |
\end{frame}}
|
|
|
577 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
578 |
*}
|
|
|
579 |
|
|
|
580 |
text_raw {*
|
|
|
581 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
582 |
\mode<presentation>{
|
|
|
583 |
\begin{frame}[c]
|
|
|
584 |
\frametitle{\LARGE A Variant of Arden's Lemma}
|
|
|
585 |
|
|
|
586 |
{\bf Arden's Lemma:}\smallskip
|
|
|
587 |
|
|
|
588 |
If \smath{[] \not\in A} then
|
|
|
589 |
\begin{center}
|
|
|
590 |
\smath{X = X; A + \text{something}}
|
|
|
591 |
\end{center}
|
|
|
592 |
has the (unique) solution
|
|
|
593 |
\begin{center}
|
|
|
594 |
\smath{X = \text{something} ; A^\star}
|
|
|
595 |
\end{center}
|
|
|
596 |
|
|
|
597 |
|
|
|
598 |
\end{frame}}
|
|
|
599 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
600 |
*}
|
|
|
601 |
|
|
|
602 |
|
|
|
603 |
text_raw {*
|
|
|
604 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
605 |
\mode<presentation>{
|
|
|
606 |
\begin{frame}<1->[t]
|
|
|
607 |
\small
|
|
|
608 |
|
|
|
609 |
\begin{center}
|
|
|
610 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
611 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
612 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
613 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
614 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
615 |
|
|
|
616 |
& & & \onslide<2->{by Arden}\\
|
|
|
617 |
|
|
|
618 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
|
619 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
620 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
621 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
622 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
623 |
|
|
|
624 |
& & & \onslide<4->{by Arden}\\
|
|
|
625 |
|
|
|
626 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
627 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
628 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
629 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
630 |
|
|
|
631 |
& & & \onslide<5->{by substitution}\\
|
|
|
632 |
|
|
|
633 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
|
634 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
635 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
636 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
637 |
|
|
|
638 |
& & & \onslide<6->{by Arden}\\
|
|
|
639 |
|
|
|
640 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
641 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
642 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
643 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
644 |
|
|
|
645 |
& & & \onslide<7->{by substitution}\\
|
|
|
646 |
|
|
|
647 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
648 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
649 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
650 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
651 |
\cdot a\cdot a^\star}}\\
|
|
|
652 |
\end{tabular}
|
|
|
653 |
\end{center}
|
|
|
654 |
|
|
|
655 |
\only<8->{
|
|
|
656 |
\begin{textblock}{6}(2.5,4)
|
|
|
657 |
\begin{block}{}
|
|
|
658 |
\begin{minipage}{8cm}\raggedright
|
|
|
659 |
|
|
|
660 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
|
|
661 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
662 |
|
|
|
663 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
664 |
|
|
|
665 |
\node[state,initial] (p_0) {$R_1$};
|
|
|
666 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
667 |
|
|
|
668 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
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|
669 |
edge [loop above] node {b} ()
|
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|
670 |
(p_1) edge [loop above] node {a} ()
|
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|
671 |
edge [bend left] node {b} (p_0);
|
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|
672 |
\end{tikzpicture}
|
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|
673 |
|
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|
674 |
\end{minipage}
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|
675 |
\end{block}
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|
676 |
\end{textblock}}
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|
677 |
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|
678 |
\end{frame}}
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679 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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680 |
*}
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681 |
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682 |
text_raw {*
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683 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
684 |
\mode<presentation>{
|
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|
685 |
\begin{frame}[c]
|
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|
686 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
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|
687 |
|
|
|
688 |
\begin{itemize}
|
|
|
689 |
\item The algorithm must terminate: Arden makes one equation smaller;
|
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|
690 |
substitution deletes one variable from the right-hand sides.\bigskip
|
|
|
691 |
|
|
|
692 |
\item We need to maintain the invariant that Arden is applicable
|
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|
693 |
(if \smath{[] \not\in A} then \ldots):\medskip
|
|
|
694 |
|
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|
695 |
\begin{center}\small
|
|
|
696 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
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|
697 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
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|
698 |
\smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
|
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|
699 |
|
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|
700 |
& & & by Arden\\
|
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|
701 |
|
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|
702 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
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|
703 |
\smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
|
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|
704 |
\end{tabular}
|
|
|
705 |
\end{center}
|
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|
706 |
|
|
|
707 |
\end{itemize}
|
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|
708 |
|
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|
709 |
|
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|
710 |
\end{frame}}
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|
711 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
712 |
*}
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713 |
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714 |
text_raw {*
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715 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
716 |
\mode<presentation>{
|
|
|
717 |
\begin{frame}[c]
|
|
|
718 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
|
|
719 |
|
|
|
720 |
\begin{itemize}
|
|
|
721 |
\item The algorithm is still a bit hairy to formalise because of our set-representation
|
|
|
722 |
for equations:
|
|
|
723 |
|
|
|
724 |
\begin{center}
|
|
|
725 |
\begin{tabular}{ll}
|
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|
726 |
\smath{\big\{ (X, \{(Y_1, r_1), (Y_2, r_2), \ldots\}),}\\
|
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|
727 |
\mbox{}\hspace{5mm}\smath{\ldots}\\
|
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|
728 |
& \smath{\big\}}
|
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|
729 |
\end{tabular}
|
|
|
730 |
\end{center}\bigskip\pause
|
|
|
731 |
|
|
|
732 |
\small
|
|
|
733 |
they are generated from \smath{U\!N\!IV /\!/ \approx_L}
|
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|
734 |
|
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|
735 |
\end{itemize}
|
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|
736 |
|
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|
737 |
|
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|
738 |
\end{frame}}
|
|
|
739 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
740 |
*}
|
|
|
741 |
|
|
|
742 |
|
|
|
743 |
|
|
|
744 |
text_raw {*
|
|
|
745 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
746 |
\mode<presentation>{
|
|
|
747 |
\begin{frame}[c]
|
|
|
748 |
\frametitle{\LARGE Other Direction}
|
|
|
749 |
|
|
|
750 |
One has to prove
|
|
|
751 |
|
|
|
752 |
\begin{center}
|
|
|
753 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
|
754 |
\end{center}
|
|
|
755 |
|
|
|
756 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
757 |
of thinking (by Chunhan), one can prove that if
|
|
|
758 |
|
|
|
759 |
\begin{center}
|
|
|
760 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
|
|
|
761 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
|
|
|
762 |
\end{center}
|
|
|
763 |
|
|
|
764 |
then
|
|
|
765 |
|
|
|
766 |
\begin{center}
|
|
|
767 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
|
|
|
768 |
\end{center}
|
|
|
769 |
|
|
|
770 |
|
|
|
771 |
|
|
|
772 |
\end{frame}}
|
|
|
773 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
774 |
*}
|
|
|
775 |
|
|
|
776 |
text_raw {*
|
|
|
777 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
778 |
\mode<presentation>{
|
|
|
779 |
\begin{frame}[c]
|
|
|
780 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
781 |
|
|
|
782 |
\begin{itemize}
|
|
|
783 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
|
|
784 |
\bigskip\pause
|
|
|
785 |
\item regular languages are closed under complementation; this is easy
|
|
|
786 |
\begin{center}
|
|
|
787 |
\smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
|
|
|
788 |
\end{center}\pause\bigskip
|
|
|
789 |
|
|
|
790 |
\item if you want to do regular expression matching (see Scott's paper)\pause\bigskip
|
|
|
791 |
|
|
|
792 |
\item I cannot yet give definite numbers
|
|
|
793 |
\end{itemize}
|
|
|
794 |
|
|
|
795 |
\only<2>{
|
|
|
796 |
\begin{textblock}{10}(4,14)
|
|
|
797 |
\small
|
|
|
798 |
\smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
|
|
|
799 |
\end{textblock}
|
|
|
800 |
}
|
|
|
801 |
|
|
|
802 |
|
|
|
803 |
|
|
|
804 |
\end{frame}}
|
|
|
805 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
806 |
*}
|
|
|
807 |
|
|
|
808 |
text_raw {*
|
|
|
809 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
810 |
\mode<presentation>{
|
|
|
811 |
\begin{frame}[c]
|
|
|
812 |
\frametitle{\LARGE Examples}
|
|
|
813 |
|
|
|
814 |
\begin{itemize}
|
|
|
815 |
\item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
|
|
|
816 |
\begin{quote}\small
|
|
|
817 |
\begin{tabular}{lcl}
|
|
|
818 |
\smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
|
|
|
819 |
\smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
|
|
|
820 |
\smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
|
|
|
821 |
\smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
|
|
|
822 |
\end{tabular}
|
|
|
823 |
\end{quote}
|
|
|
824 |
|
|
|
825 |
\item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
|
|
|
826 |
\begin{quote}\small
|
|
|
827 |
\begin{tabular}{lcl}
|
|
|
828 |
\smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
|
|
|
829 |
\smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
|
|
|
830 |
\smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
|
|
|
831 |
\smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
|
|
|
832 |
& \smath{\vdots} &\\
|
|
|
833 |
\end{tabular}
|
|
|
834 |
\end{quote}
|
|
|
835 |
\end{itemize}
|
|
|
836 |
|
|
|
837 |
\end{frame}}
|
|
|
838 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
839 |
*}
|
|
|
840 |
|
|
|
841 |
|
|
|
842 |
text_raw {*
|
|
|
843 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
844 |
\mode<presentation>{
|
|
|
845 |
\begin{frame}[c]
|
|
|
846 |
\frametitle{\LARGE What We Have Not Achieved}
|
|
|
847 |
|
|
|
848 |
\begin{itemize}
|
|
|
849 |
\item regular expressions are not good if you look for a minimal
|
|
|
850 |
one for a language (DFAs have this notion)\pause\bigskip
|
|
|
851 |
|
|
|
852 |
\item Is there anything to be said about context free languages:\medskip
|
|
|
853 |
|
|
|
854 |
\begin{quote}
|
|
|
855 |
A context free language is where every string can be recognised by
|
|
|
856 |
a pushdown automaton.
|
|
|
857 |
\end{quote}
|
|
|
858 |
\end{itemize}
|
|
|
859 |
|
|
|
860 |
\end{frame}}
|
|
|
861 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
862 |
*}
|
|
|
863 |
|
|
|
864 |
|
|
|
865 |
text_raw {*
|
|
|
866 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
867 |
\mode<presentation>{
|
|
|
868 |
\begin{frame}[c]
|
|
|
869 |
\frametitle{\LARGE Conclusion}
|
|
|
870 |
|
|
|
871 |
\begin{itemize}
|
|
|
872 |
\item on balance regular expression are superior
|
|
|
873 |
to DFAs, in my opinion\bigskip
|
|
|
874 |
|
|
|
875 |
\item I cannot think of a reason to not teach regular languages
|
|
|
876 |
to students this way (!?)\bigskip
|
|
|
877 |
|
|
|
878 |
\item I have never ever seen a proof of Myhill-Nerode based on
|
|
|
879 |
regular expressions\bigskip
|
|
|
880 |
|
|
|
881 |
\item no application, but lots of fun\bigskip
|
|
|
882 |
|
|
|
883 |
\item great source of examples
|
|
|
884 |
|
|
|
885 |
\end{itemize}
|
|
|
886 |
|
|
|
887 |
\end{frame}}
|
|
|
888 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
889 |
*}
|
|
|
890 |
|
|
|
891 |
(*<*)
|
|
|
892 |
end
|
|
|
893 |
(*>*) |