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FGI2: Blatt3, Aufgabe 1 teilweise bearbeitet

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Jim Martens
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\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{paralist}
\usepackage{gauss}
\usepackage{pgfplots}
\usepackage[locale=DE,exponent-product=\cdot,detect-all]{siunitx}
\usepackage{tikz}
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,arrows,decorations.markings}
\usepackage{polynom}
\usepackage{multirow}
\usepackage[german]{fancyref}
\polyset{style=C, div=:,vars=x}
\pgfplotsset{compat=1.8}
\pagenumbering{arabic}
% ensures that paragraphs are separated by empty lines
\parskip 12pt plus 1pt minus 1pt
\parindent 0pt
% define how the sections are rendered
\def\thesection{2.\arabic{section})}
\def\thesubsection{\arabic{subsection}.}
\def\thesubsubsection{(\roman{subsubsection})}
% some matrix magic
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
\hskip -\arraycolsep
\let\@ifnextchar\new@ifnextchar
\array{#1}}
\makeatother
\begin{document}
\author{Benjamin Kuffel, Jim Martens\\Gruppe 6}
\title{Hausaufgaben zum 3. November}
\maketitle
\setcounter{section}{3}
\section{} %3.3
\subsection{}
\begin{alignat*}{2}
L(A_{1}) &=& (a^{*}ba^{*}b)^{*}(a^{*} + a^{*}ba^{*}) \\
L(A_{2}) &=& a^{*}b(a^{*}ba^{*}b)^{*} \\
L^{\omega}(A_{1}) &=& (a^{*}ba^{*}b)^{\omega} + ((a^{*}ba^{*}b)^{*}a^{*}ba^{\omega}) \\
L^{\omega}(A_{2}) &=& (a^{*}ba^{*}b)^{\omega}
\end{alignat*}
\subsection{}
\begin{figure}
\begin{tikzpicture}[node distance=2cm]
\node[state,initial] (qs) {\((q, s)\)};
\node[state] (rt) [right=of qs] {\((r, t)\)};
\node[state,accepting] (pt) [above=of qs] {\((p, t)\)};
\path[->] (qs) edge[bend left] node [above] {\(b\)} (rt)
(qs) edge[loop below] node [below] {\(a\)} (qs)
(rt) edge[loop right] node [right] {\(a\)} (rt)
(rt) edge[bend left] node [below] {\(b\)} (qs)
(qs) edge node [left] {\(b\)} (pt)
(pt) edge[loop right] node [right] {\(a\)} (pt);
\end{tikzpicture}
\caption{initiale Zusammenhangskomponente des Produktautomaten \(A_{3}\)}
\label{fig:1}
\end{figure}
Siehe \fref{fig:1} für die Konstruktion.
\subsection{}
\begin{alignat*}{2}
L(A_{3}) &=& (a^{*}ba^{*}b)^{*}ba^{*} \\
L^{\omega}(A_{3}) &=& (a^{*}ba^{*}b)^{*}a^{*}ba^{\omega} \\
L(A_{1}) \cap L(A_{2}) &=&
\end{alignat*}
\subsection{}
\begin{figure}
\begin{tikzpicture}[node distance=2cm]
\node[state,initial,accepting] (qs1) {\((q, s, 1)\)};
\node[state] (qs2) [above=of qs1] {\((q, s, 2)\)};
\node[state] (rt1) [right=of qs1] {\((r, t, 1)\)};
\node[state,accepting] (pt1) [below=of qs1] {\((p, t, 1)\)};
\node[state] (rt2) [above=of rt1] {\((r, t, 2)\)};
\node[state] (pt2) [right=of pt1] {\((p, t, 2)\)};
\path[->] (qs1) edge[bend right] node [above left] {\(b\)} (rt2)
(qs1) edge node [right] {\(a\)} (qs2)
(qs2) edge node [above] {\(b\)} (rt2)
(qs2) edge[loop left] node [left] {\(a\)} (qs2)
(rt2) edge node [below right] {\(a\)} (rt1)
(rt2) edge[bend right] node [below right] {\(b\)} (qs1)
(rt1) edge[loop right] node [right] {\(a\)} (rt1)
(rt1) edge node [below] {\(b\)} (qs1)
(qs1) edge[bend left] node [above right] {\(b\)} (pt2)
(pt2) edge[bend left] node [below] {\(a\)} (pt1)
(pt1) edge[bend left] node [below] {\(a\)} (pt2);
\end{tikzpicture}
\caption{initiale Zusammenhangskomponente des Produktautomaten \(A_{4}\)}
\label{fig:2}
\end{figure}
Siehe \fref{fig:2} für die Konstruktion.
\subsection{}
\begin{alignat*}{2}
L(A_{4}) &=& ((aa^{*}b + b)(b + aa^{*}b))^{*}(((aa^{*}b + b)(b + aa^{*}b)) + ba(aa)^{*}) \\
L^{\omega}(A_{4}) &=& ((aa^{*}b + b)(b + aa^{*}b))^{\omega}(((aa^{*}b + b)(b + aa^{*}b)) + ba(aa)^{\omega})
\end{alignat*}
\section{} %3.4
\end{document}