Questionnaire-2024/sat_solvers_dpll/0008.tex

\item \lect Consider the following conflict graph with the following set of clauses:

\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
    thick,base node/.style={circle,draw,minimum size=20pt}, real node/.style={double,circle,draw,minimum size=20pt}]

    \node[base node] (1) {$a$};
    \node[base node] (2) [below of=1]{$\lnot b$};
    \node[base node] (3) [right of=1] {$\lnot c$};
    \node[base node] (4) [right of=2] {$d$};
    \node[base node] (5) [right of=3] {$\lnot e$};
    \node[base node] (6) [right of=4] {$e$};
    \node[base node] (7) [right of=5] {$\bot$};
    \path[]
        (1) edge [] node {$1$} (3)
        (2) edge [] node {$3$} (4)
        (4) edge [] node {$1$} (3)
        (3) edge [] node {$6$} (5)
              edge [] node {$7$} (6)
        (5) edge [] node {} (7)
        (6) edge [] node {} (7);
\end{tikzpicture}

\begin{dpllCNFInput}
  \item $\{\lnot a, \lnot c, \lnot d\}$
  \item $\{a, \lnot d\}$
  \item $\{b, d\}$
  \item $\{\lnot b, d, e\}$
  \item $\{\lnot b, \lnot e\}$
  \item $\{c, \lnot e\}$
  \item $\{c, e\}$
\end{dpllCNFInput}

Give the resolution proof for the given conflict graph and clauses and state the clause to be learned from the conflict.