Questionnaire-2024/bdds/0010_sol.tex

$$f = (r \land \lnot p) \lor (r \land \lnot p) \lor (s \land \lnot r) \lor (\lnot s \land r)\lor (r \land q),$$

\begin{cofactors}
$f_{p}$ \= $= \lnot r \lor(s \land \lnot r) \lor (\lnot s \land  r) \lor (r \land q)$ \\
\>$f_{pq}$ \= $= \true$\\
\>$f_{p\lnot q}$ \= $=  r \lor  (s \land \lnot r) \lor (\lnot s \land r)$\\
\>\>$f_{p\lnot qr}$ \= $= \true$\\
\>\>$f_{p\lnot q\lnot r}$ \= $= s$\\
\>\>\>$f_{p\lnot q\lnot rs}$ \= $= \true$\\
\>\>\>$f_{p\lnot q\lnot r\lnot s}$ \= $= \false$\\
$f_{\lnot p}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r) \lor (r \land q)$ \\
\>$f_{\lnot pq}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r)$ \\
\>$f_{\lnot p\lnot q}$ \= $= r \lor(s \land \lnot r) \lor (\lnot s \land  r)$ \\
\>\>$\Rightarrow q$ does not have an influence on the formula. These cofactors can be skipped.\\
\>$f_{\lnot pr}$ \= $= \true$ \\
\>$f_{\lnot p\lnot r}$ \= $= s = f_{p\lnot q\lnot r}$ \\
\end{cofactors}

The final ROBDD:\\
\begin{center}
\begin{bdd}[4em]
  \node[func node] (f)                      {$f$};
  \node[cofactor] (p1)  [below      of=f]   {$q$};

  \node[cofactor] (q1)  [below left of=p1]  {$q$};
  \node[cofactor] (r1)  [below right of=p1] {$r$};

  \node[cofactor] (r2)  [below right of=q1,yshift=-0.8em,xshift=-1.5em] {$r$};
  \node[phantom]  (q1L) [below left of=q1]  {};

  \node[phantom]  (r1L) [below left of=r1] {};

  \node[cofactor] (s1)  [below left of=r2] {$s$};
  \node[phantom]  (r2L) [below left of=r2]  {};
  \node[phantom]  (r2R) [below right of=r2]  {};

  \node[phantom]  (s2L) [below left of=s1]  {};
  \node[phantom]  (s2R) [below right of=s1] {};

  \funcEdge{f}{p1}

  \thenEdge{p1}{q1}
  \elseEdge{p1}{r1}

  \thenEdge{q1}{q1L}
  \negatedEdge{q1}{r2}

  \thenEdge{r2}{s1}
  \negatedEdge{r2}{r2R}

  \thenEdge{r1}{r1L}
  \elseEdge[bend left]{r1}{s1}

  \thenEdge{s1}{s2L}
  \negatedEdge{s1}{s2R}
\end{bdd}
\end{center}