Questionnaire-2024/bdds/1036_sol.tex

\begin{enumerate}
\item using \textit{variable order} $c < a < d < b$:\\
\begin{minipage}{0.5\textwidth}
\begin{tabbing}
$f$ \= $= (a \lor b \lor c) \land \lnot d$\\
	\>$ f_c $ \= $= \lnot d$ \\
	    \>\>$ f_{ca} $ \= $= \lnot d$ \\
	    \>\>$ f_{c \lnot a} $ \= $= \lnot d$ \\
	    \>\>\Rightarrow a does not have an influence\\
	    \>\>on the formula.\\
	    \>\>These cofactors can be skipped.\\
        \>\>$ f_{cd} $ \= $= \bot$ \\
        \>\>$ f_{c \lnot d} $ \= $= \top $ \\
	\>$ f_{\lnot c} $ \= $= (a \lor b) \land \lnot d$ \\
	    \>\>$ f_{\lnot ca} $ \= $= \lnot d = f_{c}$ \\
	    \>\>$ f_{\lnot c \lnot a} $ \= $= b \land \lnot d$ \\
	        \>\>\>$ f_{\lnot c \lnot ad} $ \= $= \bot$ \\
	        \>\>\>$ f_{\lnot c \lnot a \lnot d} $ \= $= b$ \\
	            \>\>\>\>$ f_{\lnot c \lnot a \lnot db} $ \= $= \top$ \\
	            \>\>\>\>$ f_{\lnot c \lnot a \lnot db} $ \= $= \bot$ \\
\end{tabbing}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{center}
\begin{forest}
for tree={circle, draw, no edge,
            minimum size=2em,
            inner sep=0pt,
            s sep=6mm,
            l sep=6mm}
[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (c.north);}
    [,phantom]
    [$c$, name=c, tikz={\draw [line] () to (d1.north); \draw [dot=.5] () to (a.north);}
        [$f_{c}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (d1.north west);}]
        [,phantom]
        [$a$, name=a, tikz={\draw [line,bend left=30] () to (d1.north); \draw [dot=.5] () to (d2.north);}
            [$d$, name=d1, tikz={\draw [line] () to (-2,-5); \draw [fulldot=.5] () to (-1,-5);}]
            [,phantom]
            [$d$, name=d2, tikz={\draw [line] () to (0.6,-5); \draw [fulldot=.5] () to (b.north);}
                [,phantom]
                [$b$, name=b, tikz={\draw [line] () to (1.3,-6.3); \draw [fulldot=.5] () to (2.3,-6.3);}]
            ]
            [$f_{\lnot c \lnot a \lnot d}$, rectangle, draw,  tikz={\draw [line] () to (b.north east);}]
        ]
        [$f_{\lnot c \lnot a}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (d2.north east);}]
    ]
    [$f_{\lnot c}$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (a.north east);}]
]
\end{forest}
\end{center}
\end{minipage}

\item
using \textit{variable order} $d < a < b < c$:\\
\begin{minipage}{0.5\textwidth}
\begin{tabbing}
$f$ \= $= (a \lor b \lor c) \land \lnot d$\\
	\>$ f_d $ \= $= \bot$ \\
	\>$ f_{\lnot d} $ \= $= a \lor b \lor c$ \\
	    \>\>$ f_{\lnot da} $ \= $= \top$ \\
	    \>\>$ f_{\lnot d \lnot a} $ \= $= b \lor c$ \\
	        \>\>\>$ f_{\lnot d \lnot ab} $ \= $= \top$ \\
	        \>\>\>$ f_{\lnot d \lnot a \lnot b} $ \= $= c$ \\
	            \>\>\>\>$ f_{\lnot d \lnot a \lnot bc} $ \= $= \top$ \\
	            \>\>\>\>$ f_{\lnot d \lnot a \lnot b \lnot c} $ \= $= \bot$ \\
\end{tabbing}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{center}
\begin{forest}
for tree={circle, draw, no edge,
            minimum size=2em,
            inner sep=0pt,
            s sep=6mm,
            l sep=6mm}
[$f$, rectangle, draw,  tikz={\draw [fulldot=.5] () to (d.north);}
    [,phantom]
    [$d$, name=d, tikz={\draw [line] () to (-0.5,-2.3); \draw [fulldot=.5] () to (a.north);}
        [,phantom]
        [,phantom]
        [$a$, name=a, tikz={\draw [line] () to (0,-3.6); \draw [dot=.5] () to (b.north);}
            [,phantom]
            [,phantom]
            [$b$, name=b, tikz={\draw [line] () to (0.5,-4.9); \draw [dot=.5] () to (c.north);}
                [,phantom]
                [,phantom]
                [$c$, name=c, tikz={\draw [line] () to (1.3,-6.2); \draw [fulldot=.5] () to (2.3,-6.2);}
                ]
                [,phantom]
            ]
            [$f_{\lnot d \lnot a \lnot b}$, rectangle, draw,  tikz={\draw [line] () to (c.north east);}]
        ]
        [$f_{\lnot d \lnot a}$, rectangle, draw,  tikz={\draw [line] () to (b.north east);}]
    ]
    [$f_{\lnot d}$, rectangle, draw,  tikz={\draw [line] () to (a.north east);}]
]
\end{forest}
\end{center}
\end{minipage}
\end{enumerate}