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Prüfungsvorbereitung.tex b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex index cdc54e8..1315d39 100644 --- a/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex +++ b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex @@ -102,9 +102,13 @@ Dazu muss $l\in V$ und $r\in \sum V\cup {\epsilon}$. \end{solution} - \part Die Menge $Reg(\sum)$ der regulären Ausdrücke über dem Alphabet ist... + \part Die Menge $Reg(\sum)$ der regulären Ausdrücke über dem Alphabet ist... \begin{solution} - + ist die kleinste Menge mit folgenden Eigenschaften: + \begin{itemize} + \item $\varnothing \in Reg(\sum), \lambda \in Reg(\sum), \sum \subseteq Reg(\sum)$ + \item Wenn $\alpha, \beta \in Reg(\sum)$, dann auch $(\alpha * \beta), (\alpha + \beta), (\alpha^*)\in Reg(\sum)$ + \end{itemize} \end{solution} \part Ein NFA ist ein Tupel $M=(...)$ @@ -121,7 +125,9 @@ \part Die von einem NFA $M=(Z,\sum,S,\delta, E)$ akzeptierte Sprache ist $L(M)=...$ (ohne Definition der Mehr-Schritt Übergangsfunktion $\delta$) \begin{solution} - + $L(M)={w\in \sum^* | \hat{\delta}(S,w)\cap E \not = \emptyset}$ + + (Das Wort wird akzeptiert wenn es mindestens einen Pfad vom Anfangs in den Endzustand gibt) \end{solution} \part Die von einem PDA $M=(Z,\sum, \Gamma, \delta, z_0, \#)$ akzeptierten Sprache ist $L(M)=...$ @@ -129,9 +135,9 @@ $L(M)=\{x\in\sum^* | \text{ es gibt } z\in Z \text{ mit } (z_0, x, \#) [...] ^*(z,\epsilon, \epsilon)\}$ \end{solution} - \part Sei $L$ eine Sprache. Für $x,y\in\sum^*$ gilt $xE_L y$ genau dann, wenn ... ($R_L$ ist die Myhill-Nerode-Äquivalenz zu L) + \part Sei $L$ eine Sprache. Für $x,y\in\sum^*$ gilt $xR_L y$ genau dann, wenn ... ($R_L$ ist die Myhill-Nerode-Äquivalenz zu L) \begin{solution} - + wenn $\forall z \in \sum^* :(xy\in L \leftrightarrow yz \in L)$ gilt \end{solution} \part Sei $M=(Z,\sum,z_0,\delta, E)$ ein DFA. Die Zustände $z,z'\in Z$ heißen erkennungsäquivalent, wenn @@ -153,22 +159,26 @@ \part Die Klasse der regulären Sprachen ist unter anderem abgeschlossen unter folgenden drei Operationen: \begin{solution} - + \begin{itemize} + \item Vereinigung $L=L_0\cup L_1$ + \item Verkettung $L=L_0L_1$ + \item Abschluss $L=L_0^*$ + \end{itemize} \end{solution} \part Sei $\sum$ ein Alphabet. Die Anzahl der Grammatiken über $\sum$ ist ... und die Anzahl der Sprachen über $\sum$ ist ... . \begin{solution} - + \end{solution} \part Unter anderem sind folgende (mind. drei) Probleme für kontextfreie Sprachen entscheidbar: \begin{solution} - + \end{solution} \part Die Klasse der Kontextfreien Sprachen ist abgeschlossen unter den Operationen 1)... und 2)... . Sie ist aber nicht abgeschlossen unter 3)... und 4)... . \begin{solution} - + \end{solution} \part Der Satz von Myhill-Nerode besagt,... @@ -210,27 +220,76 @@ \end{tikzpicture} \end{center} \begin{solution} + \begin{center} + \begin{tikzpicture}[node distance = 3cm, on grid, auto] + \node (q1) [state, initial, initial text = {}] {1}; + \node (q3) [state, accepting, right = of q1] {3}; + \node (q12) [state, above = of q1] {\{1,2\}}; + \node (q123) [state, accepting, right = of q12] {\{1,2,3\}}; + + \path [-stealth, thick] + (q1) edge node {a} (q12) + (q1) edge node {b} (q3) + (q12) edge [loop above] node {b}() + (q12) edge [bend left] node {a} (q123) + (q123) edge [bend left] node {a} (q12) + (q123) edge [loop above] node {b}() + (q3) edge node {a} (q123) + (q3) edge [loop right] node {b}() + + ; + \end{tikzpicture} + \end{center} \end{solution} \part Betrachte den folgenden NFA X. Berechne einen DFA Y mit $L(X)=L(Y)$. \begin{center} \begin{tikzpicture}[node distance = 3cm, on grid, auto] - \node (q0) [state, initial, initial text = {}] {1}; - \node (q1) [state, above right = of q0] {2}; - \node (q2) [state, accepting, below right = of q1] {3}; + \node (q1) [state, initial, initial text = {}] {1}; + \node (q2) [state, above right = of q1] {2}; + \node (q3) [state, accepting, below right = of q2] {3}; \path [-stealth, thick] - (q0) edge node {b} (q1) - (q0) edge node {a} (q2) - (q0) edge [loop above] node {b}() - (q1) edge node {a} (q2) - (q1) edge [loop above] node {a}() - (q2) edge [bend left] node {b} (q0) - (q2) edge [bend left] node {a} (q1) - (q2) edge [loop above] node {b}(); + (q1) edge node {b} (q2) + (q1) edge node {a} (q3) + (q1) edge [loop above] node {b}() + (q2) edge node {a} (q3) + (q2) edge [loop above] node {a}() + (q3) edge [bend left] node {b} (q1) + (q3) edge [bend left] node {a} (q2) + (q3) edge [loop above] node {b}(); \end{tikzpicture} \end{center} \begin{solution} + \begin{center} + \begin{tikzpicture}[node distance = 3cm, on grid, auto] + \node (q1) [state, initial, initial text = {}] {1}; + \node (q2) [state, right = of q12] {2}; + \node (q3) [state, accepting, right = of q1] {3}; + \node (q12) [state, above = of q1] {\{1,2\}}; + \node (q13) [state, accepting, right = of q3] {\{1,3\}}; + \node (q23) [state, accepting, right = of q123] {\{2,3\}}; + \node (q123) [state, accepting, above right = of q12] {\{1,2,3\}}; + + \path [-stealth, thick] + (q1) edge node {a} (q3) + (q1) edge node {b} (q12) + (q2) edge node {a} (q23) + (q2) edge [loop left] node {b}() + (q3) edge node {a} (q2) + (q3) edge node {b} (q13) + (q12) edge node {a} (q123) + (q12) edge [loop above] node {b}() + (q13) edge [bend left] node {a} (q3) + (q13) edge node {b} (q123) + (q23) edge [loop above] node {a}() + (q23) edge node {b} (q13) + (q123) edge node {a} (q23) + (q123) edge [loop above] node {b}() + + ; + \end{tikzpicture} + \end{center} \end{solution} \part Betrachte den folgenden DFA X. Berechne den minimalen DFA Y mit $L(X)=L(Y)$. @@ -258,6 +317,17 @@ \end{tikzpicture} \end{center} \begin{solution} + * erster Schritt, (*) zweiter Schritt + + \begin{tabular}{c|c|c|c|c|c} + 2 & * \\\hline + 3 & * & * \\\hline + 4 & (*) & * & * \\\hline + 5 & * & * & * & * \\\hline + 6 & (*) & * & * & (*) & * \\\hline + & 1 & 2 & 3 & 4 & 5 + \end{tabular} + \end{solution} @@ -276,7 +346,7 @@ (q1) edge node {a} (q5) (q2) edge [loop above] node {b}() (q2) edge node {a} (q3) - (q3) edge node {a} (q5) + (q3) edge [bend left] node {a} (q5) (q3) edge node {b} (q6) (q4) edge node {a} (q1) (q4) edge node {b} (q5) @@ -334,7 +404,7 @@ \begin{solution} \end{solution} - \part Seien $A\supseteq \sum^*$ und B$\supseteq \Gamma^*$. Eine Reduktion von A auf B ist ... + \part Seien $A\supseteq \sum^*$ und B$\supseteq \Gamma^*$. Eine Reduktion von A auf B ist ... \begin{solution} \end{solution} @@ -342,18 +412,18 @@ \begin{solution} \end{solution} - \part Sei $f:N\rightarrow N$ eine monotone Funktion. Die Klasse $TIME(f)$ besteht aus allen Sprachen L, für die es eine Turingmaschine $M$ gibt mit ... + \part Sei $f:N\rightarrow N$ eine monotone Funktion. Die Klasse $TIME(f)$ besteht aus allen Sprachen L, für die es eine Turingmaschine $M$ gibt mit ... \begin{solution} \end{solution} \end{parts} \question Sätze der Berechnbarkeitstheorie: Vervollständige die folgenden Aussagen \begin{parts} - \part Zu jeder Mehrband-Turingmaschine $M$ gibt es ... + \part Zu jeder Mehrband-Turingmaschine $M$ gibt es ... \begin{solution} \end{solution} - \part Sei $f:N^k\rightarrow\mathbb{N}$ eine Funktion für ein $k\in\mathbb{N}$. Die folgenden Aussagen sind äquivalent: 1) $f$ ist Turing-berechenbar, 2)..., 3)..., 4)... + \part Sei $f:N^k\rightarrow\mathbb{N}$ eine Funktion für ein $k\in\mathbb{N}$. Die folgenden Aussagen sind äquivalent: 1) $f$ ist Turing-berechenbar, 2)..., 3)..., 4)... \begin{solution} \end{solution} @@ -377,11 +447,11 @@ \begin{solution} \end{solution} - \part Gebe ein Loop Programm an, das die Funktion $f:\mathbb{N}\rightarrow \mathbb{N}$ mit $f(n_1,n_2)=2n_1n_2$ berechnet. Verwende nur elementare Anweisungen und keine Abkürzungen. + \part Gebe ein Loop Programm an, das die Funktion $f:\mathbb{N}\rightarrow \mathbb{N}$ mit $f(n_1,n_2)=2n_1n_2$ berechnet. Verwende nur elementare Anweisungen und keine Abkürzungen. \begin{solution} \end{solution} - \part Gebe ein GoTo Programm an, das die Funktion $g:\mathbb{N}\rightarrow\mathbb{N}$ mit $g(n_1,n_2)=|n_1-n_2|$ berechnet. Verwende nur elementare Anweisungen und keine Abkürzungen. + \part Gebe ein GoTo Programm an, das die Funktion $g:\mathbb{N}\rightarrow\mathbb{N}$ mit $g(n_1,n_2)=|n_1-n_2|$ berechnet. Verwende nur elementare Anweisungen und keine Abkürzungen. \begin{solution} \end{solution}

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