--- a/Slides/Slides7.thy Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1246 +0,0 @@
-(*<*)
-theory Slides7
-imports "~~/src/HOL/Library/LaTeXsugar" "Main"
-begin
-
-declare [[show_question_marks = false]]
-
-notation (latex output)
- set ("_") and
- Cons ("_::/_" [66,65] 65)
-
-(*>*)
-
-text_raw {*
- \renewcommand{\slidecaption}{Beijing, 29.~April 2011}
-
- \newcommand{\abst}[2]{#1.#2}% atom-abstraction
- \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
- \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
- \newcommand{\unit}{\langle\rangle}% unit
- \newcommand{\app}[2]{#1\,#2}% application
- \newcommand{\eqprob}{\mathrel{{\approx}?}}
- \newcommand{\freshprob}{\mathrel{\#?}}
- \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction
- \newcommand{\id}{\varepsilon}% identity substitution
-
- \newcommand{\bl}[1]{\textcolor{blue}{#1}}
- \newcommand{\gr}[1]{\textcolor{gray}{#1}}
- \newcommand{\rd}[1]{\textcolor{red}{#1}}
-
- \newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}}
- \newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}}
- \newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}}
-
- \renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont}
- \newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont}
- \newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont}
- \newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont}
-
- \newcommand{\LL}{$\mathbb{L}\,$}
-
-
- \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
- {rgb(0mm)=(0,0,0.9);
- rgb(0.9mm)=(0,0,0.7);
- rgb(1.3mm)=(0,0,0.5);
- rgb(1.4mm)=(1,1,1)}
-
- \def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex}
- \usebeamercolor[fg]{subitem projected}
- {\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}}
- \pgftext{%
- \usebeamerfont*{subitem projected}}
- \end{pgfpicture}}
-
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1>[t]
- \frametitle{%
- \begin{tabular}{@ {\hspace{-3mm}}c@ {}}
- \\
- \LARGE Verifying a Regular Expression\\[-1mm]
- \LARGE Matcher and Formal Language\\[-1mm]
- \LARGE Theory\\[5mm]
- \end{tabular}}
- \begin{center}
- Christian Urban\\
- \small Technical University of Munich, Germany
- \end{center}
-
-
- \begin{center}
- \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
- University of Science and Technology in Nanjing
- \end{center}
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{This Talk: 3 Points}
- \large
- \begin{itemize}
- \item It is easy to make mistakes.\bigskip
- \item Theorem provers can prevent mistakes, {\bf if} the problem
- is formulated so that it is suitable for theorem provers.\bigskip
- \item This re-formulation can be done, even in domains where
- we least expect it.
- \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$_1$ + r$_2$}\\
- & \bl{$\mid$} & \bl{r$_1$ $\cdot$ 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}
-
- \only<2>{
- \begin{textblock}{9}(3.6,11.8)
- \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm]
-
- \hspace{10mm}\begin{tikzpicture}
- \coordinate (m1) at (0.4,1);
- \draw (0,0.3) node (m2) {\small\color{gray}rexp};
- \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
-
- \coordinate (s1) at (0.81,1);
- \draw (1.3,0.3) node (s2) {\small\color{gray} string};
- \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
- \end{tikzpicture}
- \end{textblock}}
-
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
- \frametitle{Specification}
-
- \small
- \begin{textblock}{6}(0,3.5)
- \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
- \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\
- &\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
- &\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\
- &\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\
- &\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\
- \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\
- \rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\
- \end{tabular}
- \end{textblock}
-
- \begin{textblock}{9}(7.3,3)
- {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip}
- \includegraphics[scale=0.325]{Screen3.png}
- \end{textblock}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
- \frametitle{Version 1}
- \small
- \mbox{}\\[-8mm]\mbox{}
-
- \begin{center}\def\arraystretch{1.05}
- \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}}
- \bl{match [] []} & \bl{$=$} & \bl{true}\\
- \bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\
- \bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\
- \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
- \bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\
- \bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\
- \bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\
- \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\
- \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
- \end{tabular}
- \end{center}
-
- \begin{textblock}{9}(0.2,1.6)
- \hspace{10mm}\begin{tikzpicture}
- \coordinate (m1) at (0.44,-0.5);
- \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps};
- \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
-
- \coordinate (s1) at (0.86,-0.5);
- \draw (1.5,0.3) node (s2) {\small\color{gray} string};
- \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
- \end{tikzpicture}
- \end{textblock}
-
- \begin{textblock}{9}(2.8,11.8)
- \bl{matches$_1$ r s $\;=\;$ match [r] s}
- \end{textblock}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[c]
- \frametitle{Testing}
-
- \small
- Every good programmer should do thourough tests:
-
- \begin{center}
- \begin{tabular}{@ {\hspace{-20mm}}lcl}
- \bl{matches$_1$ (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\
- \bl{matches$_1$ (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\
- \bl{matches$_1$ (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\
- \bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\
- \bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\
- \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\
- \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\
- \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}}
- \end{tabular}
- \end{center}
-
- \onslide<3->
- {Looks OK \ldots let's ship it to customers\hspace{5mm}
- \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[c]
- \frametitle{Version 1}
-
- \only<1->{Several hours later\ldots}\pause
-
-
- \begin{center}
- \begin{tabular}{@ {\hspace{0mm}}lcl}
- \bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\
- \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\}
- \end{tabular}
- \end{center}
-
- \small
- \onslide<3->{
- \begin{center}
- \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
- \ldots\\
- \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
- \ldots\\
- \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
- \end{tabular}
- \end{center}}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
- \frametitle{Testing}
-
- \begin{itemize}
- \item While testing is an important part in the process of programming development\pause\ldots
-
- \item we can only test a {\bf finite} amount of examples.\bigskip\pause
-
- \begin{center}
- \colorbox{cream}
- {\gr{\begin{minipage}{10cm}
- ``Testing can only show the presence of errors, never their
- absence.'' (Edsger W.~Dijkstra)
- \end{minipage}}}
- \end{center}\bigskip\pause
-
- \item In a theorem prover we can establish properties that apply to
- {\bf all} input and {\bf all} output.
-
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
- \frametitle{Version 2}
- \mbox{}\\[-14mm]\mbox{}
-
- \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
-
- \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{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
- \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
-
- \bl{derivative r []} & \bl{$=$} & \bl{r} & \\
- \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
- \end{tabular}\medskip
-
- \bl{matches$_2$ r s $=$ nullable (derivative r s)}
-
- \begin{textblock}{6}(9.5,0.9)
- \begin{flushright}
- \color{gray}``if r matches []''
- \end{flushright}
- \end{textblock}
-
- \begin{textblock}{6}(9.5,6.18)
- \begin{flushright}
- \color{gray}``derivative w.r.t.~a char''
- \end{flushright}
- \end{textblock}
-
- \begin{textblock}{6}(9.5,12.1)
- \begin{flushright}
- \color{gray}``deriv.~w.r.t.~a string''
- \end{flushright}
- \end{textblock}
-
- \begin{textblock}{6}(9.5,13.98)
- \begin{flushright}
- \color{gray}``main''
- \end{flushright}
- \end{textblock}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
- \frametitle{Is the Matcher Error-Free?}
-
- We expect that
-
- \begin{center}
- \begin{tabular}{lcl}
- \bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
- \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\
- \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
- \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\
- \end{tabular}
- \end{center}
- \pause\pause\bigskip
- By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip
-
- \begin{tabular}{lrcl}
- Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\
- & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\
- \end{tabular}
-
- \only<4->{
- \begin{textblock}{3}(0.9,4.5)
- \rd{\huge$\forall$\large{}r s.}
- \end{textblock}}
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1>[c]
- \frametitle{
- \begin{tabular}{c}
- \mbox{}\\[23mm]
- \LARGE Demo
- \end{tabular}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
-
- \mbox{}\\[-2mm]
-
- \small
- \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
- \bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\
- \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\
- \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\
- \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\
- \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\
- \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\
- \end{tabular}\medskip
-
- \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
- \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\
- \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\
- \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\
- \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\
- \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\
- & & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\
- \bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\
-
- \bl{derivative r []} & \bl{$=$} & \bl{r} & \\
- \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
- \end{tabular}\medskip
-
- \bl{matches r s $=$ nullable (derivative r s)}
-
- \only<2>{
- \begin{textblock}{8}(1.5,4)
- \includegraphics[scale=0.3]{approved.png}
- \end{textblock}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{No Automata?}
-
- You might be wondering why I did not use any automata?
-
- \begin{itemize}
- \item {\bf Def.:} A \alert{regular language} is one where there is a DFA that
- recognises it.\bigskip\pause
- \end{itemize}
-
-
- There are many reasons why this is a good definition:\medskip
- \begin{itemize}
- \item pumping lemma
- \item closure properties of regular languages\\ (e.g.~closure under complement)
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{Really Bad News!}
-
- DFAs are bad news for formalisations in theorem provers. They might
- be represented as:
-
- \begin{itemize}
- \item graphs
- \item matrices
- \item partial functions
- \end{itemize}
-
- All constructions are messy to reason about.\bigskip\bigskip
- \pause
-
- \small
- \only<2>{
- Constable et al needed (on and off) 18 months for a 3-person team
- to formalise automata theory in Nuprl including Myhill-Nerode. There is
- only very little other formalised work on regular languages I know of
- in Coq, Isabelle and HOL.}
- \only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
- automata with no inaccessible states \ldots''
- }
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{}
- \large
- \begin{center}
- \begin{tabular}{p{9cm}}
- My point:\bigskip\\
-
- The theory about regular languages can be reformulated
- to be more\\ suitable for theorem proving.
- \end{tabular}
- \end{center}
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE The Myhill-Nerode Theorem}
-
- \begin{itemize}
- \item provides necessary and suf\!ficient conditions for a language
- being regular (pumping lemma only necessary)\medskip
-
- \item will help with closure properties of regular languages\bigskip\pause
-
- \item key is the equivalence relation:\smallskip
- \begin{center}
- \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
- \end{center}
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE The Myhill-Nerode Theorem}
-
- \mbox{}\\[5cm]
-
- \begin{itemize}
- \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Equivalence Classes}
-
- \begin{itemize}
- \item \smath{L = []}
- \begin{center}
- \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
- \end{center}\bigskip\bigskip
-
- \item \smath{L = [c]}
- \begin{center}
- \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
- \end{center}\bigskip\bigskip
-
- \item \smath{L = \varnothing}
- \begin{center}
- \smath{\Big\{U\!N\!IV\Big\}}
- \end{center}
-
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Regular Languages}
-
- \begin{itemize}
- \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
- such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
-
- \item Myhill-Nerode:
-
- \begin{center}
- \begin{tabular}{l}
- finite $\Rightarrow$ regular\\
- \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r.\; L = \mathbb{L}(r)}\\[3mm]
- regular $\Rightarrow$ finite\\
- \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
- \end{tabular}
- \end{center}
-
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Final Equiv.~Classes}
-
- \mbox{}\\[3cm]
-
- \begin{itemize}
- \item \smath{\text{finals}\,L \dn
- \{{\lbrack\mkern-2mu\lbrack{s}\rbrack\mkern-2mu\rbrack}_\approx\;|\; s \in L\}}\\
- \medskip
-
- \item we can prove: \smath{L = \bigcup (\text{finals}\,L)}
-
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Transitions between ECs}
-
- \smath{L = \{[c]\}}
-
- \begin{tabular}{@ {\hspace{-7mm}}cc}
- \begin{tabular}{c}
- \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
- \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
-
- %\draw[help lines] (0,0) grid (3,2);
-
- \node[state,initial] (q_0) {$R_1$};
- \node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
- \node[state] (q_2) [below right of=q_0] {$R_3$};
-
- \path[->] (q_0) edge node {c} (q_1)
- edge node [swap] {$\Sigma-{c}$} (q_2)
- (q_2) edge [loop below] node {$\Sigma$} ()
- (q_1) edge node {$\Sigma$} (q_2);
- \end{tikzpicture}
- \end{tabular}
- &
- \begin{tabular}[t]{ll}
- \\[-20mm]
- \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
-
- \smath{R_1}: & \smath{\{[]\}}\\
- \smath{R_2}: & \smath{\{[c]\}}\\
- \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
- \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}}
- \end{tabular}
-
- \end{tabular}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Systems of Equations}
-
- Inspired by a method of Brzozowski\;'64, we can build an equational system
- characterising the equivalence classes:
-
- \begin{center}
- \begin{tabular}{@ {\hspace{-20mm}}c}
- \\[-13mm]
- \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
- \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
-
- %\draw[help lines] (0,0) grid (3,2);
-
- \node[state,initial] (p_0) {$R_1$};
- \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
-
- \path[->] (p_0) edge [bend left] node {a} (p_1)
- edge [loop above] node {b} ()
- (p_1) edge [loop above] node {a} ()
- edge [bend left] node {b} (p_0);
- \end{tikzpicture}\\
- \\[-13mm]
- \end{tabular}
- \end{center}
-
- \begin{center}
- \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
- & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
- & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
- \onslide<3->{we can prove}
- & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
- & \onslide<3->{\smath{R_1;; \mathbb{L}(b) \,\cup\, R_2;;\mathbb{L}(b) \,\cup\, \{[]\}}}\\
- & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
- & \onslide<3->{\smath{R_1;; \mathbb{L}(a) \,\cup\, R_2;;\mathbb{L}(a)}}\\
- \end{tabular}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1>[t]
- \small
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
- \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
- & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
- \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
- & \onslide<1->{\smath{R_1; a + R_2; a}}\\
-
- & & & \onslide<2->{by Arden}\\
-
- \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
- & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
- \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
- & \only<2>{\smath{R_1; a + R_2; a}}%
- \only<3->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<4->{by Arden}\\
-
- \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
- & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
- \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
- & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<5->{by substitution}\\
-
- \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
- & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
- \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
- & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<6->{by Arden}\\
-
- \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
- & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
- \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
- & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<7->{by substitution}\\
-
- \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
- & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
- \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
- & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
- \cdot a\cdot a^\star}}\\
- \end{tabular}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE A Variant of Arden's Lemma}
-
- {\bf Arden's Lemma:}\smallskip
-
- If \smath{[] \not\in A} then
- \begin{center}
- \smath{X = X; A + \text{something}}
- \end{center}
- has the (unique) solution
- \begin{center}
- \smath{X = \text{something} ; A^\star}
- \end{center}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[t]
- \small
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
- \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
- & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
- \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
- & \onslide<1->{\smath{R_1; a + R_2; a}}\\
-
- & & & \onslide<2->{by Arden}\\
-
- \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
- & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
- \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
- & \only<2>{\smath{R_1; a + R_2; a}}%
- \only<3->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<4->{by Arden}\\
-
- \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
- & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
- \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
- & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<5->{by substitution}\\
-
- \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
- & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
- \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
- & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<6->{by Arden}\\
-
- \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
- & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
- \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
- & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
-
- & & & \onslide<7->{by substitution}\\
-
- \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
- & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
- \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
- & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
- \cdot a\cdot a^\star}}\\
- \end{tabular}
- \end{center}
-
- \only<8->{
- \begin{textblock}{6}(2.5,4)
- \begin{block}{}
- \begin{minipage}{8cm}\raggedright
-
- \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
- \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
-
- %\draw[help lines] (0,0) grid (3,2);
-
- \node[state,initial] (p_0) {$R_1$};
- \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
-
- \path[->] (p_0) edge [bend left] node {a} (p_1)
- edge [loop above] node {b} ()
- (p_1) edge [loop above] node {a} ()
- edge [bend left] node {b} (p_0);
- \end{tikzpicture}
-
- \end{minipage}
- \end{block}
- \end{textblock}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE The Equ's Solving Algorithm}
-
- \begin{itemize}
- \item The algorithm must terminate: Arden makes one equation smaller;
- substitution deletes one variable from the right-hand sides.\bigskip
-
- \item We need to maintain the invariant that Arden is applicable
- (if \smath{[] \not\in A} then \ldots):\medskip
-
- \begin{center}\small
- \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
- \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
- \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
-
- & & & by Arden\\
-
- \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
- \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
- \end{tabular}
- \end{center}
-
- \end{itemize}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE The Other Direction}
-
- One has to prove
-
- \begin{center}
- \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
- \end{center}
-
- by induction on \smath{r}. This is straightforward for \\the base cases:\small
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{1mm}}l}
- \smath{U\!N\!IV /\!/ \!\approx_{\emptyset}} & \smath{= \{U\!N\!IV\}}\smallskip\\
- \smath{U\!N\!IV /\!/ \!\approx_{\{[]\}}} & \smath{\subseteq \{\{[]\}, U\!N\!IV - \{[]\}\}}\smallskip\\
- \smath{U\!N\!IV /\!/ \!\approx_{\{[c]\}}} & \smath{\subseteq \{\{[]\}, \{[c]\}, U\!N\!IV - \{[], [c]\}\}}
- \end{tabular}
- \end{center}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{\LARGE The Other Direction}
-
- More complicated are the inductive cases:\\ one needs to prove that if
-
- \begin{center}
- \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{3mm}
- \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
- \end{center}
-
- then
-
- \begin{center}
- \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{\LARGE Helper Lemma}
-
- \begin{center}
- \begin{tabular}{p{10cm}}
- %If \smath{\text{finite} (f\;' A)} and \smath{f} is injective
- %(on \smath{A}),\\ then \smath{\text{finite}\,A}.
- Given two equivalence relations \smath{R_1} and \smath{R_2} with
- \smath{R_1} refining \smath{R_2} (\smath{R_1 \subseteq R_2}).\\
- Then\medskip\\
- \smath{\;\;\text{finite} (U\!N\!IV /\!/ R_1) \Rightarrow \text{finite} (U\!N\!IV /\!/ R_2)}
- \end{tabular}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\Large Derivatives and Left-Quotients}
- \small
- Work by Brozowski ('64) and Antimirov ('96):\pause\smallskip
-
-
- \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
- \multicolumn{4}{@ {}l}{Left-Quotient:}\\
- \multicolumn{4}{@ {}l}{\bl{$\text{Ders}\;\text{s}\,A \dn \{\text{s'} \;|\; \text{s @ s'} \in A\}$}}\bigskip\\
-
- \multicolumn{4}{@ {}l}{Derivative:}\\
- \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{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
- \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
-
- \bl{ders [] r} & \bl{$=$} & \bl{r} & \\
- \bl{ders (s @ [c]) r} & \bl{$=$} & \bl{der c (ders s r)} & \\
- \end{tabular}\pause
-
- \begin{center}
- \alert{$\Rightarrow$}\smath{\;\;\text{Ders}\,\text{s}\,(\mathbb{L}(\text{r})) = \mathbb{L} (\text{ders s r})}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Left-Quotients and MN-Rels}
-
- \begin{itemize}
- \item \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}\medskip
- \item \bl{$\text{Ders}\;s\,A \dn \{s' \;|\; s @ s' \in A\}$}
- \end{itemize}\bigskip
-
- \begin{center}
- \smath{x \approx_A y \Longleftrightarrow \text{Ders}\;x\;A = \text{Ders}\;y\;A}
- \end{center}\bigskip\pause\small
-
- which means
-
- \begin{center}
- \smath{x \approx_{\mathbb{L}(r)} y \Longleftrightarrow
- \mathbb{L}(\text{ders}\;x\;r) = \mathbb{L}(\text{ders}\;y\;r)}
- \end{center}\pause
-
- \hspace{8.8mm}or
- \smath{\;x \approx_{\mathbb{L}(r)} y \Longleftarrow
- \text{ders}\;x\;r = \text{ders}\;y\;r}
-
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Partial Derivatives}
-
- Antimirov: \bl{pder : rexp $\Rightarrow$ rexp set}\bigskip
-
- \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
- \bl{pder c ($\varnothing$)} & \bl{$=$} & \bl{\{$\varnothing$\}} & \\
- \bl{pder c ([])} & \bl{$=$} & \bl{\{$\varnothing$\}} & \\
- \bl{pder c (d)} & \bl{$=$} & \bl{if c = d then \{[]\} else \{$\varnothing$\}} & \\
- \bl{pder c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(pder c r$_1$) $\cup$ (pder c r$_2$)} & \\
- \bl{pder c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{(pder c r$_1$) $\odot$ r$_2$} & \\
- & & \bl{\hspace{-10mm}$\cup$ (if nullable r$_1$ then pder c r$_2$ else $\varnothing$)}\\
- \bl{pder c (r$^*$)} & \bl{$=$} & \bl{(pder c r) $\odot$ r$^*$} &\smallskip\\
- \end{tabular}
-
- \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
- \bl{pders [] r} & \bl{$=$} & \bl{r} & \\
- \bl{pders (s @ [c]) r} & \bl{$=$} & \bl{pder c (pders s r)} & \\
- \end{tabular}\pause
-
- \begin{center}
- \alert{$\Rightarrow$}\smath{\;\;\text{Ders}\,\text{s}\,(\mathbb{L}(\text{r})) = \bigcup (\mathbb{L}\;`\; (\text{pders s r}))}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{\LARGE Final Result}
-
- \mbox{}\\[7mm]
-
- \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_1}}}
- refines \bl{x $\approx_{\mathbb{L}(\text{r})}$ y}\pause
- \item \smath{\text{finite} (U\!N\!IV /\!/ R_1)} \bigskip\pause
- \item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}. Qed.
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE What Have We Achieved?}
-
- \begin{itemize}
- \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
- \bigskip\pause
- \item regular languages are closed under complementation; this is now easy\medskip
- \begin{center}
- \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
- \end{center}
- \end{itemize}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Examples}
-
- \begin{itemize}
- \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
- \begin{quote}\small
- \begin{tabular}{lcl}
- \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
- \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
- \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
- \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
- \end{tabular}
- \end{quote}
-
- \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
- \begin{quote}\small
- \begin{tabular}{lcl}
- \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
- \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
- \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
- \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
- & \smath{\vdots} &\\
- \end{tabular}
- \end{quote}
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE What We Have Not Achieved}
-
- \begin{itemize}
- \item regular expressions are not good if you look for a minimal
- one for a language (DFAs have this notion)\pause\bigskip
-
- \item Is there anything to be said about context free languages:\medskip
-
- \begin{quote}
- A context free language is where every string can be recognised by
- a pushdown automaton.\bigskip
- \end{quote}
- \end{itemize}
-
- \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Conclusion}
-
- \begin{itemize}
- \item We formalised the Myhill-Nerode theorem based on
- regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip
-
- \item Seems to be a common theme: algorithms need to be reformulated
- to better suit formal treatment.\smallskip
-
- \item The most interesting aspect is that we are able to
- implement the matcher directly inside the theorem prover
- (ongoing work).\smallskip
-
- \item Parsing is a vast field which seem to offer new results.
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1>[b]
- \frametitle{
- \begin{tabular}{c}
- \mbox{}\\[13mm]
- \alert{\LARGE Thank you very much!}\\
- \alert{\Large Questions?}
- \end{tabular}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-
-(*<*)
-end
-(*>*)
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