13ss.theoinf: notes/tex/ue02_notes.tex comparison

comparison notes/tex/ue02_notes.tex @ 44:15351d87ce76

transition notes

author	Markus Kaiser <markus.kaiser@in.tum.de>
date	Thu, 11 Jul 2013 22:06:26 +0200
parents	27fedbbdab6d
children

comparison

equal deleted inserted replaced

-:c14b92bfa07f
+:15351d87ce76
 \input{preamble.tex}
+\input{frames.tex}
 \title{Übung 2: Konversion RE $\rightarrow$ DFA}
 \subtitle{Theoretische Informatik Sommersemester 2013}
 \author{\href{mailto:markus.kaiser@in.tum.de}{Markus Kaiser}}
 \begin{document}
+\showUnit{titel}
-\begin{frame}
+\showUnit{regex}
-\titlepage
+\showUnit{rezuenfa}
-\end{frame}
+\showUnit{rezuenfazwei}
+\showUnit{enfazunfa}
-\begin{frame}
+\showUnit{nfazudfa}
-\setbeamercovered{dynamic}
+\showUnit{produktautomat}
-\frametitle{Reguläre Ausdrücke}
-\begin{definition}[Regulärer Ausdruck]
-\alert{Reguläre Ausdrücke} sind induktiv definiert
-\begin{itemize}
-\item \alert{$\emptyset$} ist ein regulärer Ausdruck
-\item \alert{$\epsilon$} ist ein regulärer Ausdruck
-\item Für alle $a \in \Sigma$ ist \alert{$a$} ein regulärer Ausdruck
-\item Sind $\alpha$ und $\beta$ reguläre Ausdrücke, dann auch
-\begin{description}
-\item[Konkatenation] \alert{$\alpha\beta$}
-\item[Veroderung] \alert{$\alpha \mid \beta$}
-\item[Wiederholung] \alert{$\alpha^*$}
-\end{description}
-\end{itemize}
-Analoge Sprachdefinition, z.b. $L(\alpha\beta) = L(\alpha)L(\beta)$
-\end{definition}
-\begin{example}
-$\alpha = (0|1)^*00$ \hfill $L(\alpha) = \left\{x \mid x \text{ Binärzahl}, x \mod 4 = 0 \right\}$
-\end{example}
-\end{frame}
-\begin{frame}[c]
-\setbeamercovered{dynamic}
-\frametitle{Konversionen}
-\begin{center}
-\begin{tikzpicture}[node distance=2cm]
-\node (nfa) {NFA};
-\node (dfa) [left of=nfa] {DFA};
-\node (enfa) [right of=nfa] {$\epsilon$-NFA};
-\node (re) [below of=nfa] {RE};
-\draw [every edge, tumred] (nfa) -- (dfa);
-\draw [every edge, tumred] (enfa) -- (nfa);
-\draw [every edge] (dfa) -- (re);
-\draw [every edge] (nfa) -- (re);
-\draw [every edge, tumred] (re) -- (enfa);
-\end{tikzpicture}
-\end{center}
-\end{frame}
-\begin{frame}
-\setbeamercovered{dynamic}
-\frametitle{RE $\rightarrow$ $\epsilon$-NFA}
-\begin{block}{Idee (Kleene)}
-Für einen Ausdruck \alert{$\gamma$} wird rekursiv mit struktureller Induktion ein $\epsilon$-NFA konstruiert.
-\end{block}
-\begin{tabu} to \linewidth {XXX}
-\alert{$\gamma = \emptyset$} & \alert{$\gamma = \epsilon$} & \alert{$\gamma = a \in \Sigma$} \\
-\begin{tikzpicture}[automaton, small, baseline=(current bounding box.north)]
-\node[state, initial] () {};
-\end{tikzpicture} &
-\begin{tikzpicture}[automaton, small, baseline=(current bounding box.north)]
-\node[state, initial, accepting] () {};
-\end{tikzpicture} &
-\begin{tikzpicture}[automaton, small, baseline=(current bounding box.north)]
-\node[state, initial] (i) {};
-\node[state, accepting] (j) [right of=i] {};
-\draw[->] (i) edge node {$a$} (j);
-\end{tikzpicture} \\
-\vspace{2em}
-\alert{$\gamma = \alpha\beta$} \\
-\multicolumn3{c}{
-\begin{tikzpicture}[automaton, small]
-\draw[tumgreen, fill=tumgreen!20] (-0.3, 1) rectangle (1.8, -1);
-\node[tumgreen] () at (0.75, -1.2) {$N_\alpha$};
-\draw[tumgreen, fill=tumgreen!20] (3.7, 1) rectangle (5.8, -1);
-\node[tumgreen] () at (4.75, -1.2) {$N_\beta$};
-\node[state, initial] (i) at (0, 0) {};
-\node[state] (j) at (1.5, 0.5) {};
-\node[state] (k) at (1.5, -0.5) {};
-\node[state] (l) at (4, 0) {};
-\node[state, accepting] (m) at (5.5, 0) {};
-\draw[->] (j) edge node {$\epsilon$} (l);
-\draw[->] (k) edge node {$\epsilon$} (l);
-\end{tikzpicture}
-}\\
-\end{tabu}
-\end{frame}
-\begin{frame}
-\setbeamercovered{dynamic}
-\frametitle{RE $\rightarrow$ $\epsilon$-NFA}
-\begin{tabu} to \linewidth {X}
-\alert{$\gamma = \alpha \mid \beta$} \\
-\centering
-\begin{tikzpicture}[automaton, small]
-\draw[tumgreen, fill=tumgreen!20] (2, 1.5) rectangle (4.5, 0.5);
-\node[tumgreen] () at (3.25, 0.3) {$N_\alpha$};
-\draw[tumgreen, fill=tumgreen!20] (2, -0.5) rectangle (4.5, -1.5);
-\node[tumgreen] () at (3.25, -1.7) {$N_\beta$};
-\node[state, initial] (i) at (0, 0) {};
-\node[state] (j) at (2.5, 1) {};
-\node[state, accepting] (k) at (4, 1) {};
-\node[state] (l) at (2.5, -1) {};
-\node[state, accepting] (m) at (4, -1) {};
-\draw[->] (i) edge node {$\epsilon$} (j);
-\draw[->] (i) edge node {$\epsilon$} (l);
-\end{tikzpicture} \\
-\vfill
-\alert{$\gamma = \alpha^*$} \\
-\centering
-\begin{tikzpicture}[automaton, small, bend angle=70]
-\draw[tumgreen, fill=tumgreen!20] (2, 1) rectangle (4.5, -1);
-\node[tumgreen] () at (3.25, -1.2) {$N_\alpha$};
-\node[state, initial, accepting] (i) at (0, 0) {};
-\node[state] (j) at (2.5, 0) {};
-\node[state, accepting] (k) at (4, 0.5) {};
-\node[state, accepting] (m) at (4, -0.5) {};
-\draw[->] (i) edge node {$\epsilon$} (j);
-\draw[->] (k) edge [bend right] node {$\epsilon$} (j);
-\draw[->] (m) edge [bend left] node[above] {$\epsilon$} (j);
-\end{tikzpicture}
-\end{tabu}
-\end{frame}
-\begin{frame}
-\setbeamercovered{dynamic}
-\frametitle{$\epsilon$-NFA $\rightarrow$ NFA}
-\begin{block}{Idee}
-Entferne $\epsilon$-Kanten durch das Bilden von $\epsilon$-Hüllen.
-\begin{enumerate}
-\item<1-> Entferne \alert{unnötige Knoten}.
-\item<1,3-> Für jeden \alert{Pfad} der Form $\epsilon\ldots\epsilon \alert{a} \epsilon\ldots\epsilon$ verbinde Anfangs- und Endknoten mit einer \alert{$a$}-Kante.
-\item<1,4-> Entferne alle \alert{$\epsilon$-Kanten} und unerreichbare Knoten.
-\item<1,5-> Wurde das leere Wort akzeptiert mache den \alert{Anfangszustand} zum Endzustand.
-\end{enumerate}
-\end{block}
-\vfill
-\begin{tikzpicture}[automaton, bend angle=40, node distance=2.1cm]
-\useasboundingbox (-1.4,2) rectangle (9, -2);
-\node<-4>[state, initial] (q0) {$q_0$};
-\node[state] (q2) [right = 3.2cm of q0] {$q_2$};
-\node[state] (q3) [right of = q2] {$q_3$};
-\node[state, accepting] (q4) [right of = q3] {$q_4$};
-\draw[->] (q2) edge node {$0$} (q3);
-\draw[->] (q3) edge node {$1$} (q4);
-\draw<1-4>[->] (q3) edge [bend right] node [above] {$\epsilon$} (q2);
-\draw[->] (q4) edge [bend right] node [above] {$1$} (q3);
-\draw<1-4>[->] (q0) edge [bend right=20] node [below] {$\epsilon$} (q4);
-\node<1>[state] (q1) [right of = q0] {$q_1$};
-\draw<1>[->] (q0) edge node {$\epsilon$} (q1);
-\draw<1>[->] (q1) edge node {$1$} (q2);
-\node<2>[state, fill=tumred!20] (q1) [right of = q0] {$q_1$};
-\draw<2>[->, tumred] (q0) edge node {$\epsilon$} (q1);
-\draw<2>[->, tumred] (q1) edge node {$0$} (q2);
-\draw<2->[->, tumblue] (q0) edge [bend left] node {$0$} (q2);
-\draw<3,4,5>[->, tumred] (q0) edge [bend right=20] node [below] {$\epsilon$} (q4);
-\draw<3>[->, tumred] (q4) edge [bend right] node [above] {$1$} (q3);
-\draw<3,4>[->, tumred] (q3) edge [bend right] node [above] {$\epsilon$} (q2);
-\draw<3->[->, tumgreen] (q0) edge node {$1$} (q2);
-\draw<4->[->, tumgreen] (q2) edge [loop above] node [above] {$0$} (q2);
-\draw<4->[->, tumgreen] (q3) edge [loop above] node [above] {$0$} (q3);
-\draw<4->[->, tumgreen] (q0) edge [bend right=20] node [above] {$1$} (q3);
-\draw<4->[->, tumgreen] (q4) edge [bend right=70] node [above] {$1$} (q2);
-\node<5>[state, initial, accepting, fill=tumgreen!20] (q0) {$q_0$};
-\node<6->[state, initial, accepting] (q0) {$q_0$};
-\end{tikzpicture}
-\end{frame}
-\begin{frame}
-\setbeamercovered{dynamic}
-\frametitle{NFA $\rightarrow$ DFA}
-\begin{block}{Idee (Potenzmengenkonstruktion)}
-Konstruiere aus einem NFA $N = (Q, \Sigma, \delta, q_0, F)$ einen DFA $D = (P(Q), \Sigma, \overline{\delta}, \{q_0\}, F_M)$ mit Zuständen aus \alert{$P(Q)$}.
-\begin{itemize}
-\item $\overline{\delta}: \alert{P(Q)} \times \Sigma \to P(Q)$ \\
-\[\overline{\delta}(S, a) := \bigcup_{q \in S} \delta(q, a)\]
-\item $F_M := \left\{S \subseteq Q \mid \alert{S \cap F} \neq \emptyset\right\}$
-\end{itemize}
-\end{block}
-\begin{tikzpicture}[automaton, bend angle=20, node distance=2.1cm]
-\useasboundingbox (-1.4,2) rectangle (9, -2);
-\node[state, initial] (q0) {$q_0$};
-\node[state, accepting] (q1) [right of = q0] {$q_1$};
-\draw[->] (q0) edge [loop above] node {$0,1$} (q0);
-\draw[->] (q0) edge node {$1$} (q1);
-\node<2->(sep) [right of = q1] {$\rightarrow$};
-\node<2->[state, initial, inner sep=1pt] (pq0) [right of = sep] {$q_{\{0\}}$};
-\node<3->[state, accepting, inner sep=0pt] (pq01) [right of = pq0] {$q_{\{0,1\}}$};
-\draw<3->[->] (pq0) edge [loop above] node {$0$} (pq0);
-\draw<3->[->] (pq0) edge [bend left] node {$1$} (pq01);
-\draw<4->[->] (pq01) edge [loop above] node {$1$} (pq01);
-\draw<4->[->] (pq01) edge [bend left] node {$0$} (pq0);
-\end{tikzpicture}
-\end{frame}
-\begin{frame}
-\setbeamercovered{dynamic}
-\frametitle{Produktautomat}
-\begin{theorem}
-Sind $M_1 = (Q_1, \Sigma, \delta_1, s_1, F_1)$ und $M_2 = (Q_2, \Sigma, \delta_2, s_2, F_2)$ DFAs, dann ist der \alert{Produkt-Automat}
-\begin{align*}
-M &:= (\alert{Q_1 \times Q_2}, \Sigma, \delta, (s_1, s_2), F_1 \times F_2) \\
-\delta\left( (q_1, q_2), a \right) &:= \left( \alert{\delta_1}(q_1, a), \alert{\delta_2}(q_2, a) \right)
-\end{align*}
-ein DFA, der $L(M_1) \cap L(M_2)$ akzeptiert.
-\end{theorem}
-\end{frame}
 \end{document}

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comparison notes/tex/ue02_notes.tex @ 44:15351d87ce76