--- a/notes/tex/ue04_notes.tex	Thu Jul 11 21:57:50 2013 +0200
+++ b/notes/tex/ue04_notes.tex	Thu Jul 11 22:06:26 2013 +0200
@@ -1,126 +1,13 @@
 \input{preamble.tex}
+\input{frames.tex}
 
 \title{Übung 4: Minimale DFAs}
 \subtitle{Theoretische Informatik Sommersemester 2013}
 \author{\href{mailto:markus.kaiser@in.tum.de}{Markus Kaiser}}
 
 \begin{document}
-
-\begin{frame}
-    \titlepage
-\end{frame}
-
-\begin{frame}
-    \frametitle{Äquivalenzen}
-    \setbeamercovered{dynamic}
-
-    \begin{definition}[Äquivalente Worte]
-        Jede Sprache $L \subseteq \Sigma^*$ induziert eine Äquivalenzrelation $\alert{\equiv_L \subseteq \Sigma^* \times \Sigma^*}$:
-        \[
-            u \alert{\equiv_L} v \Longleftrightarrow \left( \forall w \in \Sigma^*. \alert{uw} \in L \Leftrightarrow \alert{vw} \in L\right)
-        \]
-    \end{definition}
-
-    \vfill
-
-    \pause
-
-    \begin{definition}[Äquivalente Zustände]
-        Zwei Zustände im DFA $A$ sind \alert{äquivalent} wenn sie die selbe Sprache akzeptieren.
-
-        \[
-            p \alert{\equiv_A} q \Longleftrightarrow \left( \forall w \in \Sigma^*. \alert{\hat{\delta}(p, w)} \in F \Leftrightarrow \alert{\hat{\delta}(q, w)} \in F \right)
-        \]
-    \end{definition}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Unterscheidbare Zustände}
-    \setbeamercovered{dynamic}
-
-    \begin{definition}[Unterscheidbarkeit]
-        Zwei Zustände sind \alert{unterscheidbar}, wenn sie unterschiedliche Sprachen akzeptieren.
-        \[
-            p \alert{\not\equiv_A} q \Longleftrightarrow \left( \exists w \in \Sigma^*. \hat{\delta}(p, w) \alert{\in} F \wedge \hat{\delta}(q, w) \alert{\not\in} F \right)
-        \]
-    \end{definition}
-
-    \begin{theorem}
-        Sind $\delta(p, a)$ und $\delta(q, a)$ unterscheidbar, dann auch $p$ und $q$.
-    \end{theorem}
-
-    \pause
-
-    \begin{tikzpicture}[automaton, bend angle=40, node distance=2.5cm]
-        \node[state, initial] (q0) {$q_0$};
-        \node[state] (q1) [right of = q0] {$q_1$};
-        \node[state] (q2) [right of = q1] {$q_2$};
-        \node[state, accepting] (q3) [right of = q2] {$q_3$};
-
-        \draw[->] (q0) edge node {$a$} (q1);
-        \draw[->] (q0) edge [bend left] node {$b$} (q2);
-        \draw[->] (q1) edge node {$a$} (q2);
-        \draw[->] (q1) edge [bend right] node {$b$} (q3);
-        \draw[->] (q2) edge node {$a,b$} (q3);
-        \draw[->] (q3) edge [loop right] node {$a,b$} (q3);
-
-        \node<3>[state, fill=tumred!35] () at (q2) {$q_2$};
-        \node<3->[state, accepting, fill=tumgreen!35] () at (q3) {$q_3$};
-
-        \node<4>[state, fill=tumred!35] () at (q0) {$q_0$};
-        \node<4>[state, fill=tumred!35] () at (q1) {$q_1$};
-        \draw<4>[->, tumred] (q0) edge [bend left] node {$b$} (q2);
-        \draw<4>[->, tumgreen] (q1) edge [bend right] node {$b$} (q3);
-    \end{tikzpicture}
-\end{frame}
-
-\begin{frame}[t]
-    \frametitle{DFA minimieren}
-    \setbeamercovered{dynamic}
-
-    \begin{block}{Idee}
-        Erzeuge den \alert{Quotientenautomaten}.
-        \begin{enumerate}
-            \item Entferne alle von $q_0$ \alert{nicht erreichbaren} Zustände
-            \item<1, 3-> Berechne die \alert{unterscheidbaren} Zustände
-            \item<1, 6-> \alert{Kollabiere} die äquivalenten Zustände
-        \end{enumerate}
-    \end{block}
-
-    \vfill
-
-    \begin{columns}[c]<2->
-        \begin{column}{.5\textwidth}<3->
-            \begin{center}
-                \begin{tabu}to .8\textwidth{|X[c]|X[c]|X[c]|X}
-                    \multicolumn{2}{l}{0} \\ \tabucline{1-1}
-                    \alt<-4>{}{\textcolor{tumgreen}{$1/a$}} & \multicolumn{2}{l}{1} \\ \tabucline{1-2}
-                    \alt<-4>{}{\textcolor{tumgreen}{$1/a$}} &  & \multicolumn{2}{l}{2} \\ \tabucline{1-3}
-                    \alt<-3>{}{\textcolor{tumred}{$\times$}} &  \alt<-3>{}{\textcolor{tumred}{$\times$}}& \alt<-3>{} {\textcolor{tumred}{$\times$}}& 3 \\ \tabucline{1-3}
-                \end{tabu}
-            \end{center}
-        \end{column}
-        \begin{column}{.5\textwidth}
-            \begin{tikzpicture}[automaton, node distance=2.5cm]
-                \useasboundingbox (-0.5, -0.5) rectangle (2, -2);
-
-                \node[state, initial] (q0) {$q_0$};
-                \node<-5>[state] (q1) [right of = q0] {$q_1$};
-                \node<-5>[state] (q2) [below of = q0] {$q_2$};
-                \node<6>[state, fill=tumred!40] (q12) [right of = q0] {$q_{12}$};
-                \node[state, accepting] (q3) [right of = q2] {$q_3$};
-
-                \draw<-5>[->] (q0) edge node {$a$} (q1);
-                \draw<-5>[->] (q0) edge node {$b$} (q2);
-                \draw<-5>[->] (q1) edge node {$a,b$} (q3);
-                \draw<-5>[->] (q2) edge node {$a,b$} (q3);
-                \draw[->] (q3) edge [loop right] node [above] {$a,b$} (q3);
-
-                \draw<6>[->] (q12) edge node {$a,b$} (q3);
-                \draw<6>[->] (q0) edge node {$a,b$} (q12);
-            \end{tikzpicture}
-        \end{column}
-    \end{columns}
-\end{frame}
-
+\showUnit{titel}
+\showUnit{aequivalentezustaende}
+\showUnit{unterscheidbarezustande}
+\showUnit{quotientenautomat}
 \end{document}