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\setlength{\headheight}{15pt}
\lhead{{\sf Math 9850, Algebraic Systems Biology}: Homework 5}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, October 25 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

\item \textbf{Asynchronous automata}. Draw the asynchronous automaton
  the following Boolean model:
  \[
  (f_1,f_2,f_3,f_4)=({\color{Blue}x_2\And\Not{x_3}},\;\;{\color{Red}\Not{x_1}},\;\; {\color{Green}x_3\Or x_4},\;\;{\color{Purple}x_1+x_2}).
  \]
  Then, partition the nodes into strongly connected components, and
  draw the acyclic directed graph formed by collapsing the SCCs into
  single nodes.  Find the attractors and classify them by type: fixed
  point, cyclic attractor, or complex attractor. The Boolean lattice
  $B_4$ is shown below.

  \[
  \begin{tikzpicture}[scale=2]
  \tikzstyle{v} = [rectangle, draw, inner sep=2pt]
  \tikzstyle{fp} = [rectangle, draw, fill=yellow,inner sep=1pt]
  \tikzstyle{f} = [faded]
  \tikzstyle{1} = [-stealth,very thick,Blue]
  \tikzstyle{11} = [stealth-stealth,very thick,Blue]
  \tikzstyle{2} = [-stealth,very thick,Red]
  \tikzstyle{3} = [-stealth,very thick,Green]
  \tikzstyle{33} = [stealth-stealth,very thick,Green]
  \tikzstyle{4} = [-stealth,very thick,Purple]
  \tikzstyle{44} = [stealth-stealth,very thick,Purple]
  \tikzset{every loop/.style={min distance=3mm,looseness=5}}
  \tikzstyle{every node}=[font=\small]
  %%
  \begin{scope}[shift={(0,0)},shorten >= 0pt,shorten <= 0pt]
    \node (0000) [v] at (0,1) {$0000$};
    \node (0001) [v] at (2.25,2) {$0001$};
    \node (0010) [v] at (.75,2) {$0010$};
    \node (0100) [v] at (-.75,2) {$0100$};
    \node (1000) [v] at (-2.25,2) {$1000$};
    \node (0011) [v] at (3.75,3) {$0011$};
    \node (0101) [v] at (2.25,3) {$0101$};
    \node (0110) [v] at (.75,3) {$0110$};
    \node (1001) [v] at (-.75,3) {$1001$};
    \node (1010) [v] at (-2.25,3) {$1010$}; 
    \node (1100) [v] at (-3.75,3) {$1100$};
    \node (0111) [v] at (2.25,4) {$0111$};
    \node (1011) [v] at (.75,4) {$1011$};
    \node (1101) [v] at (-.75,4) {$1101$};
    \node (1110) [v] at (-2.25,4) {$1110$};
    %\draw[dashed] (-2.25,4) circle [x radius=.4cm, y radius=.2cm];
    \node (1111) [v] at (0,5) {$1111$};
    \draw [f] (0000) to (0001);
    \draw [f] (0010) to (0000);
    \draw [f] (0000) to (0100);
    \draw [f] (0000) to (1000);
    \draw [f] (0001) to (0011);
    \draw [f] (0101) to (0001);
    \draw [f] (0001) to (1001);
    \draw [f] (0010) to (0011);
    \draw [f] (0010) to (0110);
    \draw [f] (0010) to (1010);
    \draw [f] (0100) to (0101);
    \draw [f] (0100) to (0110);
    \draw [f] (0100) to (1100);
    \draw [f] (1001) to (1000);
    \draw [f] (1010) to (1000);
    \draw [f] (1000) to (1100);
    \draw [f] (0011) to (0111);
    \draw [f] (0011) to (1011);
    \draw [f] (0111) to (0101);
    \draw [f] (0101) to (1101);
    \draw [f] (0110) to (0111);
    \draw [f] (0110) to (1110);
    \draw [f] (1001) to (1011);
    \draw [f] (1101) to (1001);
    \draw [f] (1001) to (1011);
    \draw [f] (1011) to (1010);
    \draw [f] (1010) to (1110);
    \draw [f] (1101) to (1100);
    \draw [f] (1100) to (1110);
    \draw [f] (0111) to (1111);
    \draw [f] (1011) to (1111);
    \draw [f] (1111) to (1101);
    \draw [f] (1111) to (1110);
    \end{scope}
\end{tikzpicture}
\]

    \end{enumerate}

\end{document}
  
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