diff -r e31b733ace44 -r f72c82bf59e5 Slides/Slides.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slides/Slides.thy Thu Nov 25 18:54:45 2010 +0000 @@ -0,0 +1,790 @@ +(*<*) +theory Slides +imports "LaTeXsugar" +begin + +notation (latex output) + set ("_") and + Cons ("_::/_" [66,65] 65) + +(*>*) + + +text_raw {* + %\renewcommand{\slidecaption}{Cambridge, 9 November 2010} + \renewcommand{\slidecaption}{Munich, 17 November 2010} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame} + \frametitle{% + \begin{tabular}{@ {}c@ {}} + \LARGE A Formalisation of the\\[-3mm] + \LARGE Myhill-Nerode Theorem\\[-3mm] + \LARGE based on Regular Expressions\\[-3mm] + \large \onslide<2>{\alert{or, Regular Languages Done Right}}\\ + \end{tabular}} + + \begin{center} + Christian Urban + \end{center} + + + \begin{center} + 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{ + \begin{frame}[c] + \frametitle{In Most Textbooks\ldots} + + \begin{itemize} + \item A \alert{regular language} is one where there is a DFA that + recognises it.\bigskip\pause + \end{itemize} + + + I can think of three reasons why this is a good definition:\medskip + \begin{itemize} + \item string matching via DFAs (yacc) + \item pumping lemma + \item closure properties of regular languages (closed under complement) + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \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>{Alexander and Tobias: ``\ldots automata theory \ldots does not come for free \ldots''} + \only<3>{ + 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<4>{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{ + \begin{frame}[t] + \frametitle{Regular Expressions} + + \ldots are a simple datatype: + + \only<1>{ + \begin{center}\color{blue} + \begin{tabular}{rcl} + rexp & $::=$ & NULL\\ + & $\mid$ & EMPTY\\ + & $\mid$ & CHR c\\ + & $\mid$ & ALT rexp rexp\\ + & $\mid$ & SEQ rexp rexp\\ + & $\mid$ & STAR rexp + \end{tabular} + \end{center}} + \only<2->{ + \begin{center} + \begin{tabular}{rcl} + \smath{r} & \smath{::=} & \smath{0} \\ + & \smath{\mid} & \smath{[]}\\ + & \smath{\mid} & \smath{c}\\ + & \smath{\mid} & \smath{r_1 + r_2}\\ + & \smath{\mid} & \smath{r_1 \cdot r_2}\\ + & \smath{\mid} & \smath{r^\star} + \end{tabular} + \end{center}} + + \only<3->{Induction and recursion principles come for free.} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{Semantics of Rexps} + + \begin{center} + \begin{tabular}{rcl} + \smath{\mathbb{L}(0)} & \smath{=} & \smath{\varnothing}\\ + \smath{\mathbb{L}([])} & \smath{=} & \smath{\{[]\}}\\ + \smath{\mathbb{L}(c)} & \smath{=} & \smath{\{[c]\}}\\ + \smath{\mathbb{L}(r_1 + r_2)} & \smath{=} & \smath{\mathbb{L}(r_1) \cup \mathbb{L}(r_2)}\\ + \smath{\mathbb{L}(r_1 \cdot r_2)} & \smath{=} & \smath{\mathbb{L}(r_1)\; ;\; \mathbb{L} (r_2)}\\ + \smath{\mathbb{L}(r^\star)} & \smath{=} & \smath{\mathbb{L}(r)^\star} + \end{tabular} + \end{center} + + \small + \begin{center} + \begin{tabular}{rcl} + \smath{L_1 ; L_2} & \smath{\dn} & \smath{\{ s_1 @ s_2 \mid s_1 \in L_1 \wedge s_2 \in L_2\}}\bigskip\\ + \multicolumn{3}{c}{ + \smath{\infer{[] \in L^\star}{}} \hspace{10mm} + \smath{\infer{s_1 @ s_2 \in L^\star}{s_1 \in L & s_2 \in L^\star}} + } + \end{tabular} + \end{center} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\LARGE Regular Expression Matching} + + \begin{itemize} + \item Harper in JFP'99: ``Functional Pearl: Proof- Directed Debugging''\medskip + \item Yi in JFP'06: ``Educational Pearl: `Proof-Directed Debugging' revisited + for a first-order version''\medskip + \item Owens et al in JFP'09: ``Regular-expression derivatives re-examined''\bigskip\pause + + \begin{quote}\small + ``Unfortunately, regular expression derivatives have been lost in the + sands of time, and few computer scientists are aware of them.'' + \end{quote} + \end{itemize} + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + + \begin{center} + \huge\bf Demo + \end{center} + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \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{ + \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{ + \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{ + \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{ + \begin{frame}[c] + \frametitle{\LARGE Final States} + + \mbox{}\\[3cm] + + \begin{itemize} + \item \smath{\text{final}_L\,X \dn}\\ + \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L} + \smallskip + \item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}} + + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes} + + \smath{L = \{[c]\}} + + \begin{tabular}{@ {\hspace{-7mm}}cc} + \begin{tabular}{c} + \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] + \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] + + %\draw[help lines] (0,0) grid (3,2); + + \node[state,initial] (q_0) {$R_1$}; + \node[state,accepting] (q_1) [above right of=q_0] {$R_2$}; + \node[state] (q_2) [below right of=q_0] {$R_3$}; + + \path[->] (q_0) edge node {c} (q_1) + edge node [swap] {$\Sigma-{c}$} (q_2) + (q_2) edge [loop below] node {$\Sigma$} () + (q_1) edge node {$\Sigma$} (q_2); + \end{tikzpicture} + \end{tabular} + & + \begin{tabular}[t]{ll} + \\[-20mm] + \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm] + + \smath{R_1}: & \smath{\{[]\}}\\ + \smath{R_2}: & \smath{\{[c]\}}\\ + \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm] + \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}} + \end{tabular} + + \end{tabular} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \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{ + \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{ + \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{ + \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{ + \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{ + \begin{frame}[c] + \frametitle{\LARGE The Equ's Solving Algorithm} + + \begin{itemize} + \item The algorithm is still a bit hairy to formalise because of our set-representation + for equations: + + \begin{center} + \begin{tabular}{ll} + \smath{\big\{ (X, \{(Y_1, r_1), (Y_2, r_2), \ldots\}),}\\ + \mbox{}\hspace{5mm}\smath{\ldots}\\ + & \smath{\big\}} + \end{tabular} + \end{center}\bigskip\pause + + \small + they are generated from \smath{U\!N\!IV /\!/ \approx_L} + + \end{itemize} + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\LARGE Other Direction} + + One has to prove + + \begin{center} + \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} + \end{center} + + by induction on \smath{r}. Not trivial, but after a bit + of thinking (by Chunhan), one can prove that if + + \begin{center} + \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm} + \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})} + \end{center} + + then + + \begin{center} + \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})} + \end{center} + + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\LARGE What Have We Achieved?} + + \begin{itemize} + \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} + \bigskip\pause + \item regular languages are closed under complementation; this is easy + \begin{center} + \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}} + \end{center}\pause\bigskip + + \item if you want to do regular expression matching (see Scott's paper)\pause\bigskip + + \item I cannot yet give definite numbers + \end{itemize} + + \only<2>{ + \begin{textblock}{10}(4,14) + \small + \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L} + \end{textblock} + } + + + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \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{ + \begin{frame}[c] + \frametitle{\LARGE What We Have Not Achieved} + + \begin{itemize} + \item regular expressions are not good if you look for a minimal + one for a language (DFAs have this notion)\pause\bigskip + + \item Is there anything to be said about context free languages:\medskip + + \begin{quote} + A context free language is where every string can be recognised by + a pushdown automaton. + \end{quote} + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\LARGE Conclusion} + + \begin{itemize} + \item on balance regular expression are superior + to DFAs, in my opinion\bigskip + + \item I cannot think of a reason to not teach regular languages + to students this way (!?)\bigskip + + \item I have never ever seen a proof of Myhill-Nerode based on + regular expressions\bigskip + + \item no application, but lots of fun\bigskip + + \item great source of examples + + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +(*<*) +end +(*>*) \ No newline at end of file