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snwCLefsM{9=coUi{?BsT|5fumCrszxz%0vSY+-C@!KJF|>hHz{0PdU^E&u=k diff --git a/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex index f54b59a..390e33f 100644 --- a/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex +++ b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex @@ -20,6 +20,8 @@ \usepackage{hyperref} \usepackage{pgfplots} \usepackage{bussproofs} +\usepackage{tikz} +\usetikzlibrary{automata, arrows.meta, positioning} %\renewcommand{\solutiontitle}{\noindent} \SolutionEmphasis{\small} \geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} @@ -128,9 +130,9 @@ \part Sei $L\supseteq \sum^*$ eine Sprache. Dann sind äquivalent: 1) L ist regulär (d.h. wird von einem DFA akzeptiert), 2)..., 3)... \begin{solution} \begin{enumerate} - \item L ist regulär (d.h. von einem DFA akzeptiert) - \item L wird von einem NFA akzeptiert - \item L ist rechtslinear (d.h. von einer Typ-3 Grammatik erzeugt) + \item L ist regulär (d.h. von einem DFA akzeptiert) + \item L wird von einem NFA akzeptiert + \item L ist rechtslinear (d.h. von einer Typ-3 Grammatik erzeugt) \end{enumerate} \end{solution} @@ -155,10 +157,51 @@ \question Konstruktionen der Automatentheorie \begin{parts} - \part Betrachte den NFA X (Bild wird noch erstellt). Berechne einen DFA Y mit $L(X)=L(Y)$. + \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}; + + \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}(); + \end{tikzpicture} + \end{center} + \begin{solution} \end{solution} - \part Betrachte den DFA X (Bild wird noch erstellt). Berechne den minimalen DFA Y mit $L(X)=L(Y)$. + \part Betrachte den folgenden DFA X (Bild wird noch erstellt). Berechne den minimalen DFA Y mit $L(X)=L(Y)$. + \begin{center} + \begin{tikzpicture}[node distance = 3cm, on grid, auto] + \node (q1) [state, initial, initial text = {}] {1}; + \node (q2) [state, accepting, right = of q1] {2}; + \node (q3) [state, accepting, right = of q2] {3}; + \node (q4) [state, below = of q1] {4}; + \node (q5) [state, accepting, right = of q4] {5}; + \node (q6) [state, right = of q5] {6}; + + \path [-stealth, thick] + (q1) edge node {b} (q2) + (q1) edge node {a} (q4) + (q2) edge node {a} (q3) + (q2) edge node {b} (q5) + (q3) edge node {b} (q5) + (q3) edge [loop above] node {a}() + (q4) edge node {a} (q2) + (q4) edge [loop left] node {b}() + (q5) edge node {a,b} (q6) + (q6) edge node {a} (q3) + (q6) edge [loop right] node {b}(); + \end{tikzpicture} + \end{center} \begin{solution} \end{solution} \end{parts}