diff -r bef3af148df5 -r 6e5d17a808d1 Slides/Slides1.thy --- a/Slides/Slides1.thy Tue Aug 23 08:42:51 2011 +0000 +++ b/Slides/Slides1.thy Tue Aug 23 11:53:25 2011 +0000 @@ -268,7 +268,7 @@ {\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause - Do we lose anything?\pause + Infrastructure for free. Do we lose anything?\pause \begin{itemize} \item pumping lemma\pause \item closure under complementation\pause @@ -302,18 +302,7 @@ \end{center} \end{itemize} - \only<2-> - \begin{textblock}{9.9}(0.7,1.2) - \begin{block}{} - \begin{minipage}{9.4cm}\raggedright - Two directions:\smallskip\\ - - 1.) \\ - 2.) \\ - \end{minipage} - \end{block} - \end{textblock}} - + \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -328,37 +317,25 @@ \mbox{}\\[5cm] \begin{itemize} - \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} + \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}} \end{itemize} - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Regular Languages} + \only<2->{ + \begin{textblock}{11.9}(1.7,3) + \begin{block}{} + \begin{minipage}{11.4cm}\raggedright + Two directions:\medskip\\ - \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{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} - \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{minipage} + \end{block} + \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -375,10 +352,10 @@ \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} + \item \smath{\text{final}_A\,X \dn}\\ + \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_A) \;\wedge\; \forall s \in X.\; s \in A} \smallskip - \item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}} + \item we can prove: \smath{A = \bigcup \{X.\;\text{final}_A\,X\}} \end{itemize} @@ -391,7 +368,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ \begin{frame}[c] - \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes} + \frametitle{\LARGE Transitions between Eq-Classes} \smath{L = \{[c]\}} @@ -463,11 +440,6 @@ \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} @@ -537,7 +509,7 @@ \begin{frame}[c] \frametitle{\LARGE A Variant of Arden's Lemma} - {\bf Arden's Lemma:}\smallskip + {\bf ``Reversed'' Arden's Lemma:}\medskip If \smath{[] \not\in A} then \begin{center} @@ -633,38 +605,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *} -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 {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -675,22 +615,45 @@ One has to prove \begin{center} - \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} + \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 (by Chunhan), one can prove that if + of thinking, one can find a \alert{refined} relation:\bigskip + \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)})} + \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} - then - - \begin{center} - \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})} - \end{center} @@ -705,22 +668,40 @@ \frametitle{\LARGE What Have We Achieved?} \begin{itemize} - \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} + \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \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}} + \smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}} \end{center}\pause\bigskip - \item if you want to do regular expression matching (see Scott's paper)\pause\bigskip + \item non-regularity (\smath{a^nb^n}) + + \begin{quote} + \begin{minipage}{8.8cm} + \begin{block}{} + \begin{minipage}{8.6cm} + If there exists a sufficiently large set \smath{B} (for example infinite), + such that - \item I cannot yet give definite numbers + \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. + \end{minipage} + \end{block} + \end{minipage}\medskip\pause + + \small(\smath{A \dn \bigcup_i a^i}) + \end{quote} + \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} + \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A} \end{textblock} } @@ -731,30 +712,6 @@ *} - -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{ @@ -762,21 +719,48 @@ \frametitle{\LARGE Conclusion} \begin{itemize} - \item on balance regular expression are superior - to DFAs, in my opinion\bigskip + \item We have never ever seen a proof of Myhill-Nerode based on + regular expressions.\smallskip\pause - \item I cannot think of a reason to not teach regular languages - to students this way (!?)\bigskip + \item great source of examples (inductions)\smallskip\pause - \item I have never ever seen a proof of Myhill-Nerode based on - regular expressions\bigskip + \item no need to fight the theorem prover:\\ + \begin{itemize} + \item first direction (790 loc)\\ + \item second direction (400 / 390 loc)\pause + \end{itemize}\smallskip - \item no application, but lots of fun\bigskip - - \item great source of examples + \item I have \alert{\bf not} yet used it for teaching of undergraduates.\pause \end{itemize} + \only<5->{ + \begin{textblock}{13.8}(1,4) + \begin{block}{}\mbox{}\hspace{3mm} + \begin{minipage}{11cm}\raggedright + \large + + {\bf Bold Claim }\alert{(not proved!)}\medskip + + {\bf 95\%} of regular language theory can be done without + automata\medskip\\\ldots this is much more tasteful. ;o) + + \end{minipage} + \end{block} + \end{textblock}} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\mbox{}\\[2cm]\textcolor{red}{Questions?}} + + + \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}