Questionnaire-2024/equivalence_checking/practical_questions/4_7_tseitin_self.tex

\item \lect Compute the propositional formula of the following circuit and transform it into an equisatisfiable formula in CNF by apply Tseitin's encoding.
For each variable you introduce, clearly indicate which subformula of $\phi$ it represents.

\tikzstyle{branch}=[fill,shape=circle,minimum size=3pt,inner sep=0pt]
\begin{tikzpicture}[label distance=2mm]
    \node (a) at (0,0) {$a$};
    \node (b) at (0,-1) {$b$};
    \node (c) at (0,-2) {$c$};
    \node (z) at (12,-1) {$z$};

    \node (not1m) at (3,0) {};
    \node (not2m) at (3,-1) {};
    \node (not3m) at (7,-2) {};
    \node (not4m) at (7,-.5) {};
    
    \node[not gate US, draw, logic gate inputs=n] at ($(2,-0)$) (not1) {\tiny NOT};
    \node[not gate US, draw, logic gate inputs=n] at ($(2,-1)$) (not2) {\tiny NOT};
    \node[not gate US, draw, logic gate inputs=n] at ($(2,-2)$) (not3) {\tiny NOT};
    \node[and gate US, draw, logic gate inputs=nn] at ($(4,-.5)$) (and1) {\tiny AND};
    \node[not gate US, draw, logic gate inputs=n] at ($(6,-.5)$) (not4) {\tiny NOT};
    \node[or gate US, draw, logic gate inputs=nn] at ($(8,-1)$) (or1) {\tiny OR};
    \node[not gate US, draw, logic gate inputs=n] at ($(10,-1)$) (not5) {\tiny NOT};

    \draw (a.east) -- (not1.input);
    \draw (b.east) -- (not2.input);
    \draw (c.east) -- (not3.input);
    \draw (not1.output) |- (not1m.center) node[above] {$t$} |- (and1.input 1);
    \draw (not2.output) |- (not2m.center) node[below] {$u$} |- (and1.input 2);
    \draw (and1.output) -- node[above] {$v$}  (not4.input);
    \draw (not3.output) |- (not3m.center) node[below] {$x$} |- (or1.input 2);
    \draw (not4.output) |- (not4m.center) node[above] {$w$} |- (or1.input 1);
    \draw (or1.output) -- node[below] {$y$} (not5.input);
    \draw (not5.output) -- (z.west);

\end{tikzpicture}