Questionnaire-2024/sat_solvers_dpll/1017.tex

\item \self 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) {$\lnot a$};
    \node[base node] (2) [right of=1]{$\lnot b$};
    \node[base node] (4) [right of=2] {$\lnot c$};
    \node[base node] (5) [right of=4] {$\lnot e$};
    \node[base node] (6) [above right of=5] {$d$};
    \node[base node] (3) [below right of=5] {$\lnot d$};
    \node[base node] (7) [below right of=6] {$\bot$};
    \path[]
    (1) edge [] node {$3$} (2)
    edge [bend right] node {$4$} (3)
    (2) edge [] node {$5$} (4)
    edge [bend left=35] node {$1$} (6)
    (4) edge [] node {$2$} (5)
    edge [bend left=20] node {$1$} (6)
    (5) edge [] node {$4$} (3)
    (3) edge [] node {} (7)
    (6) edge [] node {} (7);
  \end{tikzpicture}

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

  State the learned clause by making a resolution proof according to the given conflict graph and given clauses.