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zdhQAEXWjqBnipk=3?g*It8FN|LW_j-xYmFjh4NO|4;p|bL`;T42UB8e5~NfPT*$AK zr9gXr7k|W+g_&;FP?rTcbyx7o<|T9NlD+LQk0I>dMccs0!q#YMG{{gnb(LoU;`Kqo z_m$ohU!M7^_#{CxLQd?3S6H-A4*VLy#VIL?4r)q<$sje^?F2eCMH?ALZ~5~;pv1>W zFh`_ko}ULvRu0LDiaG^vuR8SF{(D1i!6M>Gr&dBz z*v&1|5+yHZZaIS+HMSHn^Q=jvUW8-(&SS?v#6I29_2~)qPy|S9+&sJ-Tp&tHNo6U} Fe*o+Ukd6QV diff --git a/Automaten, Sprachen und Komplexität - Übung.tex b/Automaten, Sprachen und Komplexität - Übung.tex index 7c46f71..f64b580 100644 --- a/Automaten, Sprachen und Komplexität - Übung.tex +++ b/Automaten, Sprachen und Komplexität - Übung.tex @@ -189,8 +189,7 @@ \textit{(a) $L(G_1)=\sum^*\{a\}\sum^*\cup\sum^*\{b\}\sum^*$ für $\sum=\{a,b,c\}$} \textit{(b) $L(G_2 ) = \{ww^R \vert w \in \{a, b\}^*: \text{w startet mit einem b}\}$ Hinweis: -Für $w=w_1w_2...w_{n-1}w_n$ -sei $w^{R} = w_{n} w_{n-1} ... w_{2} w_{1}$ das umgekehrte Wort.} +Für $w=w_1w_2...w_{n-1}w_n$ sei $w^{R} = w_{n} w_{n-1} ... w_{2} w_{1}$ das umgekehrte Wort.} \textit{(c) $L(G_3)$ ist die Menge der Polynomgleichungen über den Variablen x, y. Hinweis: Ein Polynom über den Variablen x, y ist induktiv wie folgt definiert: $0, 1, x, y$ sind Polynome und falls $f,g$ Polynome sind, so auch $(f+g)$ und $(f*g)$.} @@ -369,16 +368,54 @@ Dann gilt: \section{Übung 04} %########################################## \subsection{Aufgabe 1} +\textit{Seien $u_1, u_2\in\sum$ zwei Wörter und $L\subseteq\sum$ eine Sprache. Ein trennendes Wort für die Myhill-Nerode Äquivalenz-klassen $[u_1]_L, [u_2]_L$ ist ein Wort $w\in\sum^*$, so dass $u_1 w\in L, u_2w < L$ oder umgekehrt. Bearbeiten Sie die folgenden Teilaufgaben:} +\textit{Wir betrachten die paarweise verschiedenen Myhill-Nerode Äquivalenzklassen $[\epsilon], [a], [c]$ der Sprache $L_a = \{w\in \{a, b, c\}^* | |w|_a \text{ ist gerade oder } |w| c \geq 1\}$. Geben Sie für jedes Paar von unterschiedlichen Äquivalenzklassen ein trennendes Wort an.} + +\textit{Wir betrachten die Myhill-Nerode Äquivalenzklassen der Sprache $L_b = \{0^l 10^m 10^{l +m} | l, m \in\mathbb{N}\}$. Geben Sie für $l\in\mathbb{N}, m\not= m'$ ein trennendes Wort für die Äquivalenzklassen $[0^l 10^m ]$ und $[0^l 10^m]$ an.} + %########################################## \subsection{Aufgabe 2} +\textit{Wenden Sie das in der Vorlesung vorgestellte Verfahren an, um zu entscheiden, ob die beiden dargestellten DFAs $M_1$ und $M_2$ die gleiche Sprache akzeptieren.} + +\includegraphics{Assets/ASK_uebung/u04_01.png} %########################################## \subsection{Aufgabe 3} +\textit{Wir betrachten das Universalitätsproblem: +\begin{description} + \item[Eingabe] NFA $M = (Z , \sum, S, \delta, E)$. + \item[Frage] Gilt $L(M) = \sum^*$? +\end{description} +Geben Sie ein Verfahren an, welches das Universalitätsproblem löst. Begründen Sie Ihre Antwort.} +%########################################## +\subsection{Aufgabe 4} +\textit{In Übung 2 Aufgabe 7 a) haben wir gezeigt, dass es für jeden NFA einen äquivalenten NFA mit genau einem Endzustand gibt. In dieser Aufgabe zeigen wir, dass dies für DFAs nicht der Fall ist. Bearbeiten Sie dazu folgende Teilaufgaben:} +\textit{(a) Geben Sie einen DFA M an, sodass jeder DFA $M_0$ mit $L(M_0) = L(M)$ mindestens zwei akzeptierende Zustände hat.} + +\textit{(b) Beweisen Sie, dass Ihr Automat M diese Eigenschaft hat. Hinweis: Es gibt einen Automaten M, der diese Eigenschaft und eine endliche Sprache akzeptiert.} + +%########################################## +\subsection{Aufgabe 5} +\textit{In dieser Aufgabe betrachten wir Sprachen für die ein DFA wesentlich mehr Zustände haben muss als ein NFA. Sei $n\in\mathbb{N}$. Wir betrachten die Sprache $K_n= \{w \in \{a, b\}^* | |w| \geq n$ und der n-letzte Buchstabe von w ist ein a}. + +\textit{(a) Geben Sie einen NFA mit minimaler Anzahl an Zuständen für $K_n$ an.} + +\textit{(b) Bestimmen Sie den Index der Myhill-Nerode Äquivalenz von $K_n$ , $Index(R_{K_n})$. Begründen Sie Ihre Antwort.} + +%########################################## +\subsection{Aufgabe 6} +\textit{Sei $\sum = \{a, b\}$. Geben Sie für die folgenden Sprachen jeweils eine kontextfreie Grammatik an.} + +\textit{(a) $L_a = \{a_n b_n | n\in\mathbb{N}\}$ } + +\textit{(b) $L_b = \{w\in\sum^* | |w|_a = |w|_b \}$ } + +\textit{(c) $L_c = \sum^*\backslash \{ww | w\in\sum^*\}$ } \newpage