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% define how the sections are rendered
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\def\thesubsection{\arabic{subsection}.}
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    place/.style={
        circle,
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        draw=black,
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    transitionH/.style={
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\begin{document}
\author{Benjamin Kuffel, Jim Martens\\Gruppe 6}
\title{Hausaufgaben zum 8. Dezember}
\maketitle

WICHTIG: Aufgrund der Aufgabe 8.3.1 ist diese Bearbeitung SEHR lang. Einige der Aufgaben befinden sich nach den geforderten Prozessabbildungen.

\setcounter{section}{2}
\section{} %8.3
	\subsection{}
	Die Prozesse sind auf den Abbildungen von \fref{fig:831-1} bis \fref{fig:831-18} zu sehen.
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
	\end{tikzpicture}
	\caption{Prozess 1 für 8.3.1}
	\label{fig:831-1}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=0.5 of b2] {b4};
		\node[place] (b5) [below=0.5 of b3] {b5};
		
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5);
	\end{tikzpicture}
	\caption{Prozess 2 für 8.3.1}
	\label{fig:831-2}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 3 für 8.3.1}
	\label{fig:831-3}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 4 für 8.3.1}
	\label{fig:831-4}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 5 für 8.3.1}
	\label{fig:831-5}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 6 für 8.3.1}
	\label{fig:831-6}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b7) [right=of b] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b4) edge (b)
			(b3) edge (b)
			(b) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 7 für 8.3.1}
	\label{fig:831-7}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b7) [right=of b] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b4) edge (b)
			(b2) edge (b)
			(b) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 8 für 8.3.1}
	\label{fig:831-8}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 9 für 8.3.1}
	\label{fig:831-9}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 10 für 8.3.1}
	\label{fig:831-10}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 11 für 8.3.1}
	\label{fig:831-11}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 12 für 8.3.1}
	\label{fig:831-12}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 13 für 8.3.1}
	\label{fig:831-13}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 14 für 8.3.1}
	\label{fig:831-14}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b8) [right=of b] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 15 für 8.3.1}
	\label{fig:831-15}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b8) [right=of b] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 16 für 8.3.1}
	\label{fig:831-16}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\node[transitionH] (d2) [right=of b8] {d};
		\node[place] (b9) [right=of d2] {b9};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b8)
			(b8) edge (d2)
			(d2) edge (b9);
	\end{tikzpicture}
	\caption{Prozess 17 für 8.3.1}
	\label{fig:831-17}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\node[transitionH] (d2) [right=of b8] {d};
		\node[place] (b9) [right=of d2] {b9};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b8)
			(b8) edge (d2)
			(d2) edge (b9);
	\end{tikzpicture}
	\caption{Prozess 18 für 8.3.1}
	\label{fig:831-18}
	\end{figure}
	
	\subsection{}
	Die \(<\)-Relation kann auf \fref{fig:832-l} gesehen werden. Die \(\lessdot \)-Relation kann auf \fref{fig:832-ld} gesehen werden. Die \textbf{li}-Relation kann auf \fref{fig:832-li} gesehen werden. Die \textbf{co}-Relation kann auf \fref{fig:832-co} gesehen werden.
	\begin{figure}
	\begin{tikzpicture}[node distance=2cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path[->] (b1) edge (a)
			(a) edge (b3)
			(a) edge (b4)
			(b4) edge (c)
			(b2) edge (c)
			(c) edge (b5)
			(b1) edge[bend left] (b3)
			(b1) edge[bend right] (b4)
			(b1) edge[bend right] (c)
			(b1) edge[bend left=90] (b5)
			(a) edge[bend left] (c)
			(a) edge (b5)
			(b2) edge[bend right] (b5)
			(b4) edge (b5);
	\end{tikzpicture}
	\caption{\(<\)-Relation}
	\label{fig:832-l}
	\end{figure}
	\begin{figure}
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path[->] (b1) edge (a)
			(a) edge (b3)
			(a) edge (b4)
			(b4) edge (c)
			(b2) edge (c)
			(c) edge (b5);
	\end{tikzpicture}
	\caption{\(\lessdot \)-Relation}
	\label{fig:832-ld}
	\end{figure}
	
	\begin{figure}
	\begin{tikzpicture}[node distance=2cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path (b1) edge (a)
			(a) edge (b3)
			(a) edge (b4)
			(b4) edge (c)
			(b2) edge (c)
			(c) edge (b5)
			(b1) edge[bend left] (b3)
			(b1) edge[bend right] (b4)
			(b1) edge[bend right] (c)
			(b1) edge[bend left=90] (b5)
			(a) edge[bend left] (c)
			(a) edge (b5)
			(b2) edge[bend right] (b5)
			(b4) edge (b5);
	\end{tikzpicture}
	\caption{\textbf{li}-Relation}
	\label{fig:832-li}
	\end{figure}
	
	\begin{figure}
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path (b1) edge (b2)
			(b3) edge (b5)
			(b3) edge (b2)
			(b2) edge (b4)
			(a) edge (b2)
			(b3) edge (b4)
			(b3) edge (c);
	\end{tikzpicture}
	\caption{\textbf{co}-Relation}
	\label{fig:832-co}
	\end{figure}
	
	\subsection{}
	P-Schnitt: b2, b4, b3 und T-Schnitt: a
\section{} %8.4
	Der Erreichbarkeitsgraph zu dieser Aufgabe ist auf \fref{fig:84-rg} zu sehen.

	Im Folgenden wird der Lösungsweg in natürlicher Sprache, mit dem reduzierten Graphen und Pseudocode erläutert. Dabei werden die strengen Zusammenhangskomponenten (kurz: SZKs) aus Platzgründen (siehe Erreichbarkeitsgraph) nicht vollständig angegeben, sondern mithilfe von natürlicher Sprache.
	
	Es gibt vier SZKs, wobei eine davon eine terminale SZK ist. \(C_{1}\) enthält alle Markierungen, die von der Startmarkierung aus durch beliebiges Schalten der Transitionen \(a,c,d\) erreicht werden können. Durch das Schalten von \(b\) wird diese SZK verlassen und man gelangt in \(C_{2}\). Diese SZK enthält wiederum alle Markierungen, die von einer ihrer Markierungen mit beliebigem Schalten von \(a,c,d\) erreicht werden können. Erneut wird die SZK durch das Schalten von \(b\) verlassen und man befindet sich in \(C_{3}\). Für diese SZK gilt die gleiche Regel, wie für die vorangegangenen. Verlässt man diese SZK über das Schalten von \(b\) gelangt man in die terminale SZK \(C_{4}\).
	
	Der reduzierte Graph ist im Vergleich zum Erreichbarkeitsgraph recht simpel. Er kann auf \fref{fig:84-rgc} gefunden werden.
	
	Nachdem nun die Vorarbeit getan ist, werden die vier Transitionen nacheinander durchgegangen, um zu zeigen, dass sie entweder lebendig sind oder nicht.
	
	Transition \(a\):
	\begin{verbatim}
		C_i = C_0
		C_i.contains(m, a) = true
		C_i = C_1
		C_i.contains(m, a) = true
		C_i = C_2
		C_i.contains(m, a) = true
		C_i = C_3
		C_i.contains(m, a) = true
		return true
	\end{verbatim}
	
	Transition \(b\):
	\begin{verbatim}
		C_i = C_0
		C_i.contains(m, b) = true
		C_i = C_1
		C_i.contains(m, b) = true
		C_i = C_2
		C_i.contains(m, b) = true
		C_i = C_3
		C_i.contains(m, b) = false
		return false
	\end{verbatim}
	
	Transition \(c\):
	\begin{verbatim}
		C_i = C_0
		C_i.contains(m, c) = true
		C_i = C_1
		C_i.contains(m, c) = true
		C_i = C_2
		C_i.contains(m, c) = true
		C_i = C_3
		C_i.contains(m, c) = true
		return true
	\end{verbatim}
	
	Transition \(d\):
	\begin{verbatim}
		C_i = C_0
		C_i.contains(m, d) = true
		C_i = C_1
		C_i.contains(m, d) = true
		C_i = C_2
		C_i.contains(m, d) = true
		C_i = C_3
		C_i.contains(m, d) = true
		return true
	\end{verbatim}
	
	Mithilfe des Algorithmus 7.3 aus dem Skript (Entscheiden einer Lebendigkeits-Invarianzeigenschaft) war es möglich zu zeigen, dass die Transitionen \(a,c\) und \(d\) lebendig sind, während Transition \(b\) dies nicht ist.
	
	\begin{figure}
	\begin{tikzpicture}[node distance=1cm]
		\node[ellipse] (m0) {2p1};
		\node[ellipse] (m1) [below=of m0] {p1 + 3p3};
		\node[ellipse] (m3) [below=of m1] {p1 + p2 + 2p3};
		\node[ellipse] (m2) [left=2 of m1] {6p3};
		\node[ellipse] (m4) [below=of m2] {p2 + 5p3};
		\node[ellipse] (m5) [below=of m3] {p1 + 2p3};
		\node[ellipse] (m6) [below=of m4] {2p2 + 4p3};
		\node[ellipse] (m7) [below=of m5] {5p3};
		\node[ellipse] (m8) [below=of m6] {3p2 + 3p3};
		\node[ellipse] (m9) [below=of m7] {p2 + 4p3};
		\node[ellipse] (m10) [below=of m9] {2p2 + 3p3};
		\node[ellipse] (m11) [below=of m8] {p1 + 2p2 + p3};
		\node[ellipse] (m12) [left=of m8] {4p2 + 2p3};
		\node[ellipse] (m13) [below=of m11] {p1 + p2 + p3};		
		\node[ellipse] (m14) [below=of m10] {3p2 + 2p3};
		\node[ellipse] (m15) [below=of m12] {p1 + 3p2};
		\node[ellipse] (m16) [below=of m13] {p1 + 2p2};
		\node[ellipse] (m17) [left=of m16] {p1 + p3};
		\node[ellipse] (m18) [below=of m16] {p1 + p2};
		\node[ellipse] (m19) [below=of m17] {4p3};
		\node[ellipse] (m20) [below=of m14] {p1};
		\node[ellipse] (m21) [below=of m18] {p2 + 3p3};
		\node[ellipse] (m22) [below=of m20] {3p3};
		\node[ellipse] (m23) [below=of m22] {p2 + 2p3};
		\node[ellipse] (m24) [below=of m21] {2p2 + 2p3};
		\path[->] (m0) edge node[right] {a} (m1)
			(m1) edge node[above] {a} (m2)
			(m1) edge node[above right] {d} (m3)
			(m2) edge node[left] {d} (m4)
			(m3) edge node[above] {a} (m4)
			(m3) edge node[right] {b} (m5)
			(m3) edge[bend right=45] node[right] {c} (m0)
			(m4) edge node[below right] {c} (m1)
			(m4) edge node[left] {d} (m6)
			(m5) edge node[right] {a} (m7)
			(m6) edge node[below right] {c} (m3)
			(m6) edge node[left] {d} (m8)
			(m7) edge node[right] {d} (m9)
			(m9) edge[bend right] node[right] {c} (m5)
			(m9) edge node[right] {d} (m10)
			(m8) edge node[left] {c} (m11)
			(m8) edge node[above left] {d} (m12)
			(m10) edge node[below right] {c} (m13)
			(m10) edge node[right] {d} (m14)
			(m11) edge node[left] {b} (m13)
			(m11) edge[bend right=45] node[right] {a} (m6)
			(m12) edge node[left] {c} (m15)
			(m15) edge node[below right] {a} (m8)
			(m15) edge node[left] {b} (m16)
			(m13) edge node[above left] {a} (m9)
			(m14) edge node[below left] {c} (m16)
			(m13) edge node[above left] {b} (m17)
			(m16) edge node[left] {b} (m18)
			(m16) edge node[below right] {a} (m10)
			(m17) edge node[left] {a} (m19)
			(m18) edge node[above] {b} (m20)
			(m18) edge node[right] {a} (m21)
			(m20) edge node[right] {a} (m22)
			(m22) edge node[right] {d} (m23)
			(m23) edge[bend right] node[right] {c} (m20)
			(m19) edge node[below left] {d} (m21)
			(m21) edge node[above right] {c} (m17)
			(m21) edge node[right] {d} (m24)
			(m24) edge[bend right=45] node[right] {c} (m18);
	\end{tikzpicture}
	\caption{Erreichbarkeitsgraph für 8.4}
	\label{fig:84-rg}
	\end{figure}

	\begin{figure}
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (c0) {\(C_{0}\)};
		\node[place] (c1) [right=of c0] {\(C_{1}\)};
		\node[place] (c2) [below=of c1] {\(C_{2}\)};
		\node[place] (c3) [left=of c2] {\(C_{3}\)};
		\path[->] (c0) edge node[above] {b} (c1)
			(c1) edge node[right] {b} (c2)
			(c2) edge node[above] {b} (c3);
	\end{tikzpicture}
	\caption{Reduzierter Erreichbarkeitsgraph aus 8.4}
	\label{fig:84-rgc}
	\end{figure}
\section{} %8.5
	\subsection{}
	Diese Transitionen sind nebenläufig, können also unabhängig voneinander schalten.
	\subsection{}
	Diese Stellen sind Teil eines Stellenschnittes, einer erreichbaren Markierung.
	\subsection{}
	Jeder Prozess ist ein Kausalnetz mit einem Abbildungspaar, welches die Bedingungen auf Stellen und die Ereignisse auf Transitionen abbildet. Das zu einem Prozess gehörende Kausalnetz ist vorgängerendlich. Es gibt jedoch auch Kausalnetze, die nicht vorgängerendlich und damit nicht Bestandteil von Prozessen sind.
	
	Der strukturelle Unterschied besteht darin, dass Kausalnetze aus Bedingungen und Ereignissen bestehen, wohingegen Prozesse ein Kausalnetz referenzieren und eine Abbildung von Bedingungen und Ereignissen auf Plätze und Transitionen enthalten. Naiv gesehen könnte man sagen, dass Kausalnetze weniger umfassen als Prozesse.
	
	Rein zahlenmäßig ist das Verhältnis jedoch umgekehrt, wie bereits mit der Vorgängerendlichkeit beschrieben.
	\subsection{}
	Das Kausalnetz passt zu den Petrinetzen \(N_{8.4a}\) und \(N_{8.4b}\). Die Abbildung für das erste Petrinetz sieht so aus:
	\begin{alignat*}{2}
		\phi(b1) &=& p1 \\
		\phi(b2) &=& p1 \\
		\phi(x) &=& a \\
		\phi(b3) &=& p1 \\
		\phi(b4) &=& p1 \\
		\phi(y) &=& b \\
		\phi(b5) &=& p1 \\
		\phi(b6) &=& p1
	\end{alignat*}
	
	Für das zweite Petrinetz sieht die Abbildung so aus:
	\begin{alignat*}{2}
		\phi(b1) &=& q2 \\
		\phi(b2) &=& q1 \\
		\phi(x) &=& s \\
		\phi(b3) &=& q3 \\
		\phi(b4) &=& q2 \\
		\phi(y) &=& t \\
		\phi(b5) &=& q3 \\
		\phi(b6) &=& q4
	\end{alignat*}
\end{document}