\item \self Find a \textit{symbolic encoding} for the \textit{transition relation} of the following \textit{Kripke structure} and
simplify your formulas. The labels of the states symbolize their symbolic encoding.
E.g. "110" = "$x_2 \land x_1 \land \lnot x_0$", i.e. the least significant bit corresponds to $x_0$,
the second-least significant bit to $x_1$, and so forth.

\begin{center}
\vspace{-2em}
\begin{tikzpicture}[auto, node distance=2cm,shorten >=1pt,
thick,node/.style={circle,draw,minimum size=25pt}]
    \node[node] (s0) {$000$};
    \node[node] (s1) [above right of=s0] {$001$};
    \node[node] (s2) [right of=s1] {$010$};
    \node[node] (s3) [below right of=s2] {$011$};
    \node[node] (s4) [below of=s3] {$100$};
    \node[node] (s5) [below left of=s4] {$101$};
    \node[node] (s6) [left of=s5] {$110$};
    \node[node] (s7) [above left of=s6] {$111$};



    \path[->] (s0) edge (s1);
    \path[->] (s1) edge (s2);
    \path[<->] (s2) edge (s3);
    \path[->] (s3) edge (s4);
    \path[->] (s4) edge (s7);
    \path[->] (s7) edge (s6);
    \path[<->] (s6) edge (s5);
    \path[->] (s3) edge (s6);
    \path[->] (s5) edge (s0);
    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
    \path[->] (s4.30) edge[bend right=90, looseness=15, out=240, in=300] (s4.60);
\end{tikzpicture}
\end{center}