Mercurial > 13ss.theoinf
diff notes/tex/ue02_notes.tex @ 44:15351d87ce76
transition notes
author | Markus Kaiser <markus.kaiser@in.tum.de> |
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date | Thu, 11 Jul 2013 22:06:26 +0200 |
parents | 27fedbbdab6d |
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--- a/notes/tex/ue02_notes.tex Thu Jul 11 21:57:50 2013 +0200 +++ b/notes/tex/ue02_notes.tex Thu Jul 11 22:06:26 2013 +0200 @@ -1,259 +1,16 @@ \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} - -\begin{frame} - \titlepage -\end{frame} - -\begin{frame} - \setbeamercovered{dynamic} - \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} - +\showUnit{titel} +\showUnit{regex} +\showUnit{rezuenfa} +\showUnit{rezuenfazwei} +\showUnit{enfazunfa} +\showUnit{nfazudfa} +\showUnit{produktautomat} \end{document}