diff -r e31b733ace44 -r f72c82bf59e5 Slides/Slides1.thy --- a/Slides/Slides1.thy Thu Nov 18 11:39:17 2010 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,699 +0,0 @@ -(*<*) -theory Slides1 -imports "LaTeXsugar" -begin - -notation (latex output) - set ("_") and - Cons ("_::/_" [66,65] 65) - -(*>*) - - -text_raw {* - \renewcommand{\slidecaption}{Cambridge, 9 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 Textbooks\ldots} - - \begin{itemize} - \item A \alert{regular language} is one where there is DFA that - recognises it.\pause - \item Pumping lemma, closure properties of regular languages (closed - under ``negation'') etc are all described and proved in terms of DFAs.\pause - - \item Similarly the Myhill-Nerode theorem, which gives necessary and sufficient - conditions for a language being regular (also describes a minimal DFA for a language). - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[t] - \frametitle{Really Bad News!} - - This is bad news for formalisations in theorem provers. DFAs might - be represented as - - \begin{itemize} - \item graphs - \item matrices - \item partial functions - \end{itemize} - - All constructions are difficult 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{ - \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 will help with closure properties of regular languages and - with the pumping lemma.\medskip - - \item provides necessary and suf\!ficient conditions for a language being - regular\bigskip\pause - - \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 {$\Sigma$} (); - \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<3->{by Arden}\\ - - \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}} - & \onslide<3->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ - \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}} - & \onslide<3->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<4->{by substitution}\\ - - \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} - & \onslide<4->{\smath{R_1; a\cdot a^\star \cdot 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 Arden}\\ - - \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ - \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<6->{by substitution}\\ - - \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{\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:} - - 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<3->{by Arden}\\ - - \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}} - & \onslide<3->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ - \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}} - & \onslide<3->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<4->{by substitution}\\ - - \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} - & \onslide<4->{\smath{R_1; a\cdot a^\star \cdot 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 Arden}\\ - - \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ - \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<6->{by substitution}\\ - - \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{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star - \cdot a\cdot a^\star}}\\ - - & & & \onslide<7->{\alert{solved form}}\\ - \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 This 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}\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 `inversion' - \begin{center} - \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}} - \end{center}\pause\bigskip - - \item regular expressions are not good if you look for a minimal - one of a language (DFA have this notion)\pause\bigskip - - \item if you want to do regular expression matching (see Scott's paper) - \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 Conclusion} - - \begin{itemize} - \item on balance regular expression are superior to DFAs\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 a lot of fun\bigskip - - \item great source of examples - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -(*<*) -end -(*>*) \ No newline at end of file