# HG changeset patch # User Christian Urban <christian dot urban at kcl dot ac dot uk> # Date 1380716252 -3600 # Node ID 15b8fc34cb082fcf9c01499aa08e890a5c8eea51 # Parent 796de251332ce111b65421b168ed65ad6b5712e5 added new slides diff -r 796de251332c -r 15b8fc34cb08 Slides/Slides6.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slides/Slides6.thy Wed Oct 02 13:17:32 2013 +0100 @@ -0,0 +1,1434 @@ +(*<*) +theory Slides5 +imports "~~/src/HOL/Library/LaTeXsugar" +begin + +notation (latex output) + set ("_") and + Cons ("_::/_" [66,65] 65) + +(*>*) + + +text_raw {* + \renewcommand{\slidecaption}{London, 29 August 2012} + \newcommand{\bl}[1]{\textcolor{blue}{#1}} + \newcommand{\sout}[1]{\tikz[baseline=(X.base), inner sep=-0.1pt, outer sep=0pt] + \node [cross out,red, ultra thick, draw] (X) {\textcolor{black}{#1}};} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame} + \frametitle{% + \begin{tabular}{@ {}c@ {}} + \\[-3mm] + \LARGE The Myhill-Nerode Theorem\\[-3mm] + \LARGE in a Theorem Prover\\[0mm] + \end{tabular}} + + \begin{center} + Christian Urban\\ + \small King's College London + \end{center}\bigskip + + \begin{center} + \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA + University of Science and Technology in Nanjing + \end{center} + + \only<2->{ + \begin{textblock}{6}(9,5.3) + \alert{\bf Isabelle/HOL} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}<1->[c] + \frametitle{} + + \mbox{}\\[2mm] + \begin{itemize} + \item my background is in + \begin{itemize} + \item \normalsize programming languages and theorem provers + \item \normalsize develop Nominal Isabelle + \end{itemize}\bigskip\bigskip\bigskip\bigskip\bigskip + + \item<1->to formalise and mechanically check proofs from + programming language research, TCS \textcolor{gray}{and OS}\bigskip + + \item<2->we found out that the variable convention can lead to + faulty proofs\ldots + \end{itemize} + + \onslide<2->{ + \begin{center} + \begin{block}{} + \color{gray} + \footnotesize + {\bf\mbox{}\hspace{-1.5mm}Variable Convention:}\\[0mm] + If $M_1,\ldots,M_n$ occur in a certain mathematical context + (e.g. definition, proof), then in these terms all bound variables + are chosen to be different from the free variables.\hfill Henk Barendregt + \end{block} + \end{center}} + + + \only<1->{ + \begin{textblock}{6}(10.9,3.5) + \includegraphics[scale=0.23]{isabelle1.png} + \end{textblock}} + + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{} + + \begin{tabular}{c@ {\hspace{2mm}}c} + \\[6mm] + \begin{tabular}{c} + \includegraphics[scale=0.11]{harper.jpg}\\[-2mm] + {\footnotesize Bob Harper}\\[-2.5mm] + {\footnotesize (CMU)} + \end{tabular} + \begin{tabular}{c} + \includegraphics[scale=0.37]{pfenning.jpg}\\[-2mm] + {\footnotesize Frank Pfenning}\\[-2.5mm] + {\footnotesize (CMU)} + \end{tabular} & + + \begin{tabular}{p{6cm}} + \raggedright + \color{gray}{published a proof on LF in\\ {\bf ACM Transactions on Computational Logic}, 2005, + $\sim$31pp} + \end{tabular}\\ + + \pause + \\[0mm] + + \begin{tabular}{c} + \includegraphics[scale=0.36]{appel.jpg}\\[-2mm] + {\footnotesize Andrew Appel}\\[-2.5mm] + {\footnotesize (Princeton)} + \end{tabular} & + + \begin{tabular}{p{6cm}} + \raggedright + \color{gray}{relied on their proof in a\\ {\bf security} critical application\\ (proof-carrying code)} + \end{tabular} + \end{tabular} + + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + + +text {* + \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex] + \tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick, + draw=black!50, top color=white, bottom color=black!20] + \tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick, + draw=red!70, top color=white, bottom color=red!50!black!20] + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}<2->[squeeze] + \frametitle{} + + \begin{columns} + + \begin{column}{0.8\textwidth} + \begin{textblock}{0}(1,2) + + \begin{tikzpicture} + \matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm] + { \&[-10mm] + \node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \& + \node (proof1) [node1] {\large Proof}; \& + \node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\ + + \onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \& + \onslide<4->{\node (def2) [node2] {\large Spec$^\text{+ex}$};} \& + \onslide<4->{\node (proof2) [node1] {\large Proof};} \& + \onslide<4->{\node (alg2) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\ + + \onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \& + \onslide<5->{\node (def3) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \& + \onslide<5->{\node (proof3) [node1] {\large Proof};} \& + \onslide<5->{\node (alg3) [node2] {\large Alg$^\text{-ex}$};} \\ + + \onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \& + \onslide<6->{\node (def4) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \& + \onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \& + \onslide<6->{\node (alg4) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\ + }; + + \draw[->,black!50,line width=2mm] (proof1) -- (def1); + \draw[->,black!50,line width=2mm] (proof1) -- (alg1); + + \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);} + \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);} + + \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);} + \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);} + + \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);} + \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);} + + \onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);} + \end{tikzpicture} + + \end{textblock} + \end{column} + \end{columns} + + + \begin{textblock}{3}(12,3.6) + \onslide<4->{ + \begin{tikzpicture} + \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h}; + \end{tikzpicture}} + \end{textblock} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{} + + \begin{itemize} + \item I also found fixable errors in my Ph.D.-thesis about cut-elimination + (examined by Henk Barendregt and Andy Pitts)\bigskip + \item found flaws in a proof about a classic OS scheduling algorithm + --- helped us to implement\\ it correctly and ef$\!$ficiently\\ + {\small\textcolor{gray}{(the existing literature ``proved'' + correct an incorrect algorithm; used in the Mars Pathfinder mission)}} + \end{itemize}\bigskip\bigskip\pause + + + {\bf Conclusion:}\smallskip + + Pencil-and-paper proofs in TCS are not foolproof, + not even expertproof. + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[t] + + \small Scott Aaronson (Berkeley/MIT):\\[-7mm]\mbox{} + \begin{center} + \begin{block}{} + \color{gray} + \small + ``I still remember having to grade hundreds of exams where the + students started out by assuming what had to be proved, or filled + page after page with gibberish in the hope that, somewhere in the + mess, they might accidentally have said something + correct. \ldots{}innumerable examples of ``parrot proofs'' --- + NP-completeness reductions done in the wrong direction, arguments + that look more like LSD trips than coherent chains of logic \ldots{}'' + \end{block} + \end{center}\pause + + \begin{tabular}{@ {}c@ {}} + Tobias Nipkow calls this the ``London Underground Phenomenon'': + \end{tabular} + + \begin{center} + \begin{tabular}{ccc} + students & \;\;\raisebox{-8mm}{\includegraphics[scale=0.16]{gap.jpg}}\;\; & proofs + \end{tabular} + \end{center} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{} + + \begin{textblock}{12.9}(1.5,2.0) + \begin{block}{} + \begin{minipage}{12.4cm}\raggedright + \large {\bf Motivation:}\\[2mm]I want to teach \alert{students} with + theorem\\ provers (especially for inductions). + \end{minipage} + \end{block} + \end{textblock}\pause + + \mbox{}\\[35mm]\mbox{} + + \begin{itemize} + \item \only<2>{\smath{\text{fib}}, \smath{\text{even}} and \smath{\text{odd}}}% + \only<3->{\sout{\smath{\text{fib}}, \smath{\text{even}} and \smath{\text{odd}}}}\medskip + \item<3-> formal language theory \\ + \mbox{}\;\;@{text "\<Rightarrow>"} nice textbooks: Kozen, Hopcroft \& Ullman\ldots + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}<1->[t] + \frametitle{Regular Expressions} + + \begin{textblock}{6}(2,4) + \begin{tabular}{@ {}rrl} + \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\ + & \bl{$\mid$} & \bl{[]}\\ + & \bl{$\mid$} & \bl{c}\\ + & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\ + & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\ + & \bl{$\mid$} & \bl{r$^*$}\\ + \end{tabular} + \end{textblock} + + \begin{textblock}{6}(8,3.5) + \includegraphics[scale=0.35]{Screen1.png} + \end{textblock} + + \begin{textblock}{6}(10.2,2.8) + \footnotesize Isabelle: + \end{textblock} + + \begin{textblock}{6}(7,12) + \footnotesize\textcolor{gray}{students have seen them and can be motivated about them} + \end{textblock} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}<1->[t] + + \mbox{}\\[-2mm] + + \small + \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} + \bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\ + \bl{nullable ([])} & \bl{$=$} & \bl{true} &\\ + \bl{nullable (c)} & \bl{$=$} & \bl{false} &\\ + \bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) $\vee$ (nullable r$_2$)} & \\ + \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) $\wedge$ (nullable r$_2$)} & \\ + \bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\ + \end{tabular}\medskip\pause + + \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} + \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\ + \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\ + \bl{der c (d)} & \bl{$=$} & \bl{if c $=$ d then [] else $\varnothing$} & \\ + \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ + \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$) + } & \\ + & & \bl{\hspace{3mm}(if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\ + \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ (r$^*$)} &\smallskip\\\pause + + \bl{derivative [] r} & \bl{$=$} & \bl{r} & \\ + \bl{derivative (c::s) r} & \bl{$=$} & \bl{derivative s (der c r)} & \\ + \end{tabular}\medskip + + \bl{matches r s $=$ nullable (derivative s r)} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE Regular Expression Matching\\[-2mm] in Education} + + \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\bigskip\bigskip\pause + \item Owens et al in JFP'09: ``Regular-expression derivatives re-examined'' + \bigskip + + \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}[t] + \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}} + \mbox{}\\[-15mm]\mbox{} + + \begin{center} + \huge\bf\textcolor{gray}{in Theorem Provers}\\ + \footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots} + \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 [very thick, red] (C) to [bend left=45] (B); + \draw [very thick, red] (D) to [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}{@ {\hspace{-5mm}}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:\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->{\underline{A solution}:\;\;use \smath{\text{nat}}s \;@{text "\<Rightarrow>"}\; state nodes\medskip} + + \only<7->{You have to \alert{rename} states!} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[t] + \frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}} + \mbox{}\\[-15mm]\mbox{} + + \begin{center} + \huge\bf\textcolor{gray}{in Theorem Provers}\\ + \footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots} + \end{center} + + \begin{itemize} + \item Kozen's paper-proof of Myhill-Nerode:\\ + requires absence of \alert{inaccessible states} + \item complementation of automata only works for \alert{complete} automata + (need sink states)\medskip + \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{} + \mbox{}\\[25mm]\mbox{} + + \begin{textblock}{13.9}(0.7,1.2) + \begin{block}{} + \begin{minipage}{13.4cm}\raggedright + {\bf Definition:}\smallskip\\ + + A language \smath{A} is \alert{regular}, provided there exists a\\ + \alert{regular expression} that matches all strings of \smath{A}. + \end{minipage} + \end{block} + \end{textblock}\pause + + {\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause + + Infrastructure for free. But do we lose anything?\medskip\pause + + \begin{minipage}{1.1\textwidth} + \begin{itemize} + \item pumping lemma\pause + \item closure under complementation\pause + \item \only<6>{regular expression matching}% + \only<7->{\sout{regular expression matching} + {\footnotesize(@{text "\<Rightarrow>"}Brzozowski'64, Owens et al '09)}} + \item<8-> most textbooks are about automata + \end{itemize} + \end{minipage} + + + \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\\ \textcolor{gray}{(pumping lemma only necessary)}\bigskip + + \item key is the equivalence relation:\medskip + \begin{center} + \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A} + \end{center} + \end{itemize} + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE The Myhill-Nerode Theorem} + + \begin{center} + \only<1>{% + \begin{tikzpicture}[scale=3] + \draw[very thick] (0.5,0.5) circle (.6cm); + \end{tikzpicture}}% + \only<2->{% + \begin{tikzpicture}[scale=3] + \draw[very thick] (0.5,0.5) circle (.6cm); + \clip[draw] (0.5,0.5) circle (.6cm); + \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4); + \end{tikzpicture}} + \end{center} + + \begin{itemize} + \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}} + \end{itemize} + + \begin{textblock}{5}(2.1,5.3) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm] + {$U\!N\!IV$}; + \draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}}; + \end{tikzpicture} + \end{textblock} + + \only<2->{% + \begin{textblock}{5}(9.1,7.2) + \begin{tikzpicture} + \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm] + {@{text "\<lbrakk>s\<rbrakk>"}$_{\approx_{A}}$}; + \draw (0.9,-1.1) node {\begin{tabular}{l}an equivalence class\end{tabular}}; + \end{tikzpicture} + \end{textblock}} + + \only<3->{ + \begin{textblock}{11.9}(1.7,3) + \begin{block}{} + \begin{minipage}{11.4cm}\raggedright + Two directions:\medskip\\ + \begin{tabular}{@ {}ll} + 1.)\;finite $\Rightarrow$ regular\\ + \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm] + 2.)\;regular $\Rightarrow$ finite\\ + \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{{\cal L}(r)})} + \end{tabular} + + \end{minipage} + \end{block} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE Initial and Final {\sout{\textcolor{gray}{States}}}} + + \begin{textblock}{8}(10, 2) + \textcolor{black}{Equivalence Classes} + \end{textblock} + + + \begin{center} + \begin{tikzpicture}[scale=3] + \draw[very thick] (0.5,0.5) circle (.6cm); + \clip[draw] (0.5,0.5) circle (.6cm); + \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4); + \only<2->{\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);} + \only<3->{\draw[red, fill] (0.2, 0.2) rectangle (0.4, 0.4); + \draw[red, fill] (0.4, 0.8) rectangle (0.6, 1.0); + \draw[red, fill] (0.6, 0.0) rectangle (0.8, 0.2); + \draw[red, fill] (0.8, 0.4) rectangle (1.0, 0.6);} + \end{tikzpicture} + \end{center} + + \begin{itemize} + \item \smath{\text{finals}\,A\,\dn \{[\!|s|\!]_{\approx_{A}}\;|\;s \in A\}} + \smallskip + \item we can prove: \smath{A = \bigcup \text{finals}\,A} + \end{itemize} + + \only<2->{% + \begin{textblock}{5}(2.1,4.6) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=blue,text=white, minimum height=2cm] + {$[] \in X$}; + \end{tikzpicture} + \end{textblock}} + + \only<3->{% + \begin{textblock}{5}(10,7.4) + \begin{tikzpicture} + \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm] + {a final}; + \end{tikzpicture} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}<-1>[c] + \frametitle{\begin{tabular}{@ {}l}\LARGE% + Transitions between Eq-Classes\end{tabular}} + + \begin{center} + \begin{tikzpicture}[scale=3] + \draw[very thick] (0.5,0.5) circle (.6cm); + \clip[draw] (0.5,0.5) circle (.6cm); + \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4); + \draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8); + \draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6); + \draw[white] (0.1,0.7) node (X) {$X$}; + \draw[white] (0.9,0.5) node (Y) {$Y$}; + \draw[blue, ->, line width = 2mm, bend left=45] (X) -- (Y); + \node [inner sep=1pt,label=above:\textcolor{blue}{$c$}] at ($ (X)!.5!(Y) $) {}; + \end{tikzpicture} + \end{center} + + \begin{center} + \smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y} + \end{center} + + \onslide<8>{ + \begin{tabular}{c} + \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] + \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] + \node[state,initial] (q_0) {$R_1$}; + \end{tikzpicture} + \end{tabular}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE Systems of Equations} + + Inspired by a method of Brzozowski\;'64:\bigskip\bigskip + + \begin{center} + \begin{tabular}{@ {\hspace{-20mm}}c} + \\[-13mm] + \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] + \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] + + %\draw[help lines] (0,0) grid (3,2); + + \node[state,initial] (p_0) {$X_1$}; + \node[state,accepting] (p_1) [right of=q_0] {$X_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{X_1} & \smath{=} & \smath{X_1;b + X_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\ + & \smath{X_2} & \smath{=} & \smath{X_1;a + X_2;a}\medskip\\ + \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{X_1}} & \onslide<1->{\smath{=}} + & \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\ + \onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}} + & \onslide<1->{\smath{X_1; a + X_2; a}}\\ + + & & & \onslide<2->{by Arden}\\ + + \onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}} + & \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\ + \onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}} + & \only<2->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<4->{by Arden}\\ + + \onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}} + & \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\ + \onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}} + & \onslide<4->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<5->{by substitution}\\ + + \onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}} + & \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ + \onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}} + & \onslide<5->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<6->{by Arden}\\ + + \onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}} + & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ + \onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}} + & \onslide<6->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<7->{by substitution}\\ + + \onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}} + & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ + \onslide<7->{\smath{X_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) {$X_1$}; + \node[state,accepting] (p_1) [right of=q_0] {$X_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}} + + \only<1,2>{% + \begin{textblock}{3}(0.6,1.2) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<2>{% + \begin{textblock}{3}(0.6,3.6) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<4>{% + \begin{textblock}{3}(0.6,2.9) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<4>{% + \begin{textblock}{3}(0.6,5.3) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<5>{% + \begin{textblock}{3}(1.0,5.6) + \begin{tikzpicture} + \node at (0,0) (A) {}; + \node at (0,1) (B) {}; + \draw[<-, line width=2mm, red] (B) to (A); + \end{tikzpicture} + \end{textblock}} + \only<5,6>{% + \begin{textblock}{3}(0.6,7.7) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<6>{% + \begin{textblock}{3}(0.6,10.1) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<7>{% + \begin{textblock}{3}(1.0,10.3) + \begin{tikzpicture} + \node at (0,0) (A) {}; + \node at (0,1) (B) {}; + \draw[->, line width=2mm, red] (B) to (A); + \end{tikzpicture} + \end{textblock}} + + \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}<1-2,4->[t] + \small + + \begin{center} + \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} + \onslide<1->{\smath{X_1}} & \onslide<1->{\smath{=}} + & \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\ + \onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}} + & \onslide<1->{\smath{X_1; a + X_2; a}}\\ + + & & & \onslide<2->{by Arden}\\ + + \onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}} + & \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\ + \onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}} + & \only<2->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<4->{by Arden}\\ + + \onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}} + & \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\ + \onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}} + & \onslide<4->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<5->{by substitution}\\ + + \onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}} + & \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ + \onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}} + & \onslide<5->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<6->{by Arden}\\ + + \onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}} + & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ + \onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}} + & \onslide<6->{\smath{X_1; a\cdot a^\star}}\\ + + & & & \onslide<7->{by substitution}\\ + + \onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}} + & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ + \onslide<7->{\smath{X_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) {$X_1$}; + \node[state,accepting] (p_1) [right of=q_0] {$X_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}} + + \only<1,2>{% + \begin{textblock}{3}(0.6,1.2) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<2>{% + \begin{textblock}{3}(0.6,3.6) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<4>{% + \begin{textblock}{3}(0.6,2.9) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<4>{% + \begin{textblock}{3}(0.6,5.3) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<5>{% + \begin{textblock}{3}(1.0,5.6) + \begin{tikzpicture} + \node at (0,0) (A) {}; + \node at (0,1) (B) {}; + \draw[<-, line width=2mm, red] (B) to (A); + \end{tikzpicture} + \end{textblock}} + \only<5,6>{% + \begin{textblock}{3}(0.6,7.7) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<6>{% + \begin{textblock}{3}(0.6,10.1) + \begin{tikzpicture} + \node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm] + {\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock}} + \only<7>{% + \begin{textblock}{3}(1.0,10.3) + \begin{tikzpicture} + \node at (0,0) (A) {}; + \node at (0,1) (B) {}; + \draw[->, line width=2mm, red] (B) to (A); + \end{tikzpicture} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE The Other Direction} + + One has to prove + + \begin{center} + \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})} + \end{center} + + by induction on \smath{r}. Not trivial, but after a bit + of thinking, one can find a \alert{refined} relation:\bigskip + + + \begin{center} + \mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c} + \begin{tikzpicture}[scale=1.1] + %Circle + \draw[thick] (0,0) circle (1.1); + \end{tikzpicture} + & + \begin{tikzpicture}[scale=1.1] + %Circle + \draw[thick] (0,0) circle (1.1); + %Main rays + \foreach \a in {0, 90,...,359} + \draw[very thick] (0, 0) -- (\a:1.1); + \foreach \a / \l in {45/1, 135/2, 225/3, 315/4} + \draw (\a: 0.65) node {\small$a_\l$}; + \end{tikzpicture} + & + \begin{tikzpicture}[scale=1.1] + %Circle + \draw[red, thick] (0,0) circle (1.1); + %Main rays + \foreach \a in {0, 45,...,359} + \draw[red, very thick] (0, 0) -- (\a:1.1); + \foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2} + \draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}}; + \end{tikzpicture}\\ + \small\smath{U\!N\!IV} & + \small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} & + \small\smath{U\!N\!IV /\!/ \alert{R}} + \end{tabular}} + \end{center} + + \begin{textblock}{5}(9.8,2.6) + \begin{tikzpicture} + \node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}}; + \end{tikzpicture} + \end{textblock} + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[t] + \frametitle{\LARGE\begin{tabular}{c}Derivatives of RExps\end{tabular}} + + \begin{itemize} + \item introduced by Brzozowski~'64 + \item produces a regular expression after a character has been ``parsed''\\[-18mm]\mbox{} + \end{itemize} + + \only<1->{% + \textcolor{blue}{% + \begin{center} + \begin{tabular}{@ {}lc@ {\hspace{3mm}}l@ {}} + der c $\varnothing$ & $\dn$ & $\varnothing$\\ + der c [] & $\dn$ & $\varnothing$\\ + der c d & $\dn$ & if c $=$ d then [] else $\varnothing$\\ + der c ($r_1 + r_2$) & $\dn$ & (der c $r_1$) $+$ (der c $r_2$)\\ + der c ($r^*$) & $\dn$ & (der c $r$) $\cdot$ ($r^*$)\\ + der c ($r_1 \cdot r_2$) & $\dn$ & ((der c $r_1$) $\cdot$ $r_2$) +\\ + & & \hspace{-3mm}(if nullable $r_1$ then der c $r_2$ else $\varnothing$)\\ + \end{tabular} + \end{center}}} + + \only<2->{ + \begin{textblock}{13}(1.5,5.7) + \begin{block}{} + \begin{quote} + \begin{minipage}{13cm}\raggedright + derivatives refine \smath{x \approx_{{\cal{L}}(r)} y}\bigskip + \begin{center} + \only<2>{\mbox{\hspace{-22mm}}\smath{{\cal{L}}(\text{ders}~x~r) = {\cal{L}}(\text{ders}~y~r) + \Longleftrightarrow x \approx_{{\cal{L}}(r)} y}} + \only<3>{\mbox{\hspace{-22mm}}\smath{\text{ders}~x~r = \text{ders}~y~r + \Longrightarrow x \approx_{{\cal{L}}(r)} y}} + \end{center}\bigskip + \ + \smath{\text{finite}(\text{ders}~A~r)}, but only modulo ACI + + \begin{center} + \begin{tabular}{@ {\hspace{-10mm}}rcl} + \smath{(r_1 + r_2) + r_3} & \smath{\equiv} & \smath{r_1 + (r_2 + r_3)}\\ + \smath{r_1 + r_2} & \smath{\equiv} & \smath{r_2 + r_1}\\ + \smath{r + r} & \smath{\equiv} & \smath{r}\\ + \end{tabular} + \end{center} + \end{minipage} + \end{quote} + \end{block} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}<2>[t] + \frametitle{\LARGE\begin{tabular}{c}Partial Derivatives of RExps\end{tabular}} + + + \only<2>{% + \textcolor{blue}{% + \begin{center} + \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} + pder c $\varnothing$ & $\dn$ & \alert{$\{\}$}\\ + pder c [] & $\dn$ & \alert{$\{\}$}\\ + pder c d & $\dn$ & if c $=$ d then $\{$[]$\}$ else $\{\}$\\ + pder c ($r_1 + r_2$) & $\dn$ & (pder c $r_1$) \alert{$\cup$} (der c $r_2$)\\ + pder c ($r^\star$) & $\dn$ & (pder c $r$) $\cdot$ $r^\star$\\ + pder c ($r_1 \cdot r_2$) & $\dn$ & (pder c $r_1$) $\cdot$ $r_2$ \alert{$\cup$}\\ + & & \hspace{-4mm}if nullable $r_1$ then (pder c $r_2$) else $\varnothing$\\ + \end{tabular} + \end{center}}} + + \only<2>{ + \begin{textblock}{6}(8.5,2.7) + \begin{block}{} + \begin{quote} + \begin{minipage}{6cm}\raggedright + \begin{itemize} + \item partial derivatives + \item by Antimirov~'95 + \end{itemize} + \end{minipage} + \end{quote} + \end{block} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[t] + \frametitle{\LARGE Partial Derivatives} + + \mbox{}\\[0mm]\mbox{} + + \begin{itemize} + + \item \alt<1>{\smath{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}} + {\smath{\underbrace{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}_{R}}} + refines \textcolor{blue}{$x$ $\approx_{{\cal L}(r)}$ $y$}\\[16mm]\pause + \item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause + \item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed. + \end{itemize} + + \only<2->{% + \begin{textblock}{5}(3.9,7.2) + \begin{tikzpicture} + \node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}}; + \draw (2.2,0) node {Antimirov '95}; + \end{tikzpicture} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[t] + \frametitle{\LARGE What Have We Achieved?} + + \begin{itemize} + \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}} + \medskip\pause + \item regular languages are closed under complementation; this is now easy + \begin{center} + \smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}} + \end{center}\pause\medskip + + \item non-regularity (\smath{a^nb^n})\medskip\pause\pause + + \item take \alert{\bf any} language\\ build the language of substrings\\ + \pause + + then this language \alert{\bf is} regular\;\; (\smath{a^nb^n} $\Rightarrow$ \smath{a^\star{}b^\star}) + + \end{itemize} + +\only<2>{ +\begin{textblock}{10}(4,14) +\small +\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A} +\end{textblock}} + +\only<4>{ +\begin{textblock}{5}(2,8.6) +\begin{minipage}{8.8cm} +\begin{block}{} +\begin{minipage}{8.6cm} +If there exists a sufficiently large set \smath{B} (for example infinitely large), +such that + +\begin{center} +\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}. +\end{center} + +then \smath{A} is not regular.\hspace{1.3cm}\small(\smath{B \dn \bigcup_n a^n}) +\end{minipage} +\end{block} +\end{minipage} +\end{textblock} +} + + \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 from 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\\ ($>$7000 loc) + \item Mirkin's partial derivative automaton construction (10600 loc) + \end{itemize} + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE Conclusion} + + \begin{itemize} + \item we have never seen a proof of Myhill-Nerode based on + regular expressions only\smallskip\pause + + \item great source of examples (inductions)\smallskip\pause + + \item no need to fight the theorem prover:\\ + \begin{itemize} + \item first direction (790 loc)\\ + \item second direction (400 / 390 loc) + \end{itemize} + + \item I am not saying automata are bad; just formal proofs about + them are quite dif$\!$ficult\pause\bigskip\medskip + + \item parsing with derivatives of grammars\\ (Matt Might ICFP'11) + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[c] + \frametitle{\LARGE An Apology} + + \begin{itemize} + \item This should all of course be done co-inductively + \end{itemize} + + \footnotesize + \begin{tabular}{@ {\hspace{4mm}}l} + From: Jasmin Christian Blanchette\\ + To: isabelle-dev@mailbroy.informatik.tu-muenchen.de\\ + Subject: [isabelle-dev] NEWS\\ + Date: \alert{\bf Tue, 28 Aug 2012} 17:40:55 +0200\\ + \\ + * {\bf HOL/Codatatype}: New (co)datatype package with support for mixed,\\ + nested recursion and interesting non-free datatypes.\\ + \\ + * HOL/Ordinals\_and\_Cardinals: Theories of ordinals and cardinals\\ + (supersedes the AFP entry of the same name).\\[2mm] + Kudos to Andrei and Dmitriy!\\ + \\ + Jasmin\\[-1mm] + ------------------------------------\\ + isabelle-dev mailing list\\ + isabelle-dev@in.tum.de\\ + \end{tabular} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode<presentation>{ + \begin{frame}[b] + \frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you very much!\\[5mm]Questions?}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +(*<*) +end +(*>*) \ No newline at end of file