Questionnaire-2024/sat_solvers_dpll/0011_graphs.tex

First Conflict in Step 6: \\
\begin{conflictgraph}
  \node[base node] (notA)                       {$\lnot a$};
  \node[base node] (notB) [below of=notA]       {$\lnot b$};
  \node[base node] (C)    [right of=notB]       {$c$};
  \node[base node] (notE) [above right of=C]    {$\lnot e$};
  \node[base node] (D)    [above right of=notE] {$d$};
  \node[base node] (notD) [below right of=notE] {$\lnot d$};
  \node[base node] (bot)  [below right of=D] {$\bot$};
  \path[]
      (notA) edge [] node {$6$} (notE)
      (notB) edge [] node {$2$} (C)
      (C)    edge [] node {$6$} (notE)
      (notE) edge [] node {$4$} (D)
      (notE) edge [] node {$5$} (notD)
      (notD) edge [] node {}    (bot)
      (D)    edge [] node {}    (bot);
\end{conflictgraph}
\begin{prooftree}
\AxiomC{$4. \; \lnot d \lor e$}
\AxiomC{$5. \; d \lor e$}
	\BinaryInfC{$e$}
	\AxiomC{$6. \; a \lor \lnot c \lor \lnot e$}
		\BinaryInfC{$a \lor \lnot c$}
		\AxiomC{$2. \; b \lor c$}
			\BinaryInfC{$ a \lor b$}
\end{prooftree}
\vspace{1cm}

Second Conflict in Step 11: \\
\begin{conflictgraph}
  \node[base node] (notA)                       {$\lnot a$};
  \node[base node] (B)    [right of=notA]       {$b$};
  \node[base node] (notD) [right of=B]          {$\lnot d$};
  \node[base node] (E)    [right of=notD]       {$e$};
  \node[base node] (C)    [below of=E]          {$c$};
  \node[base node] (notE) [right of=C]          {$\lnot e$};
  \node[base node] (bot)  [above right of=notE] {$\bot$};
  \path[]
      (notA) edge [] node {$8$} (B)
      (B)    edge [] node {$3$} (notD)
      (notD) edge [] node {$5$} (E)
      (notD) edge [] node {$7$} (C)
      (C)    edge [] node {$6$} (notE)
      (notE) edge [] node {}    (bot)
      (E)    edge [] node {}    (bot);
\end{conflictgraph}


\begin{prooftree}
\AxiomC{$6. \; a \lor \lnot c \lor \lnot e$}
\AxiomC{$7. \; \lnot b \lor c \lor d$}
	\BinaryInfC{$a \lor \lnot b \lor d \lor \lnot e$}
	\AxiomC{$5. \; d \lor e$}
	  \BinaryInfC{$a \lor \lnot b \lor d$}
		\AxiomC{$3. \; \lnot b \lor \lnot d$}
			\BinaryInfC{$ a \lor \lnot b$}
		  \AxiomC{$8. \; a \lor b$}
			  \BinaryInfC{$a$}
\end{prooftree}
\vspace{1cm}

Second Conflict in Step 16: \\
\begin{conflictgraph}
  \node (0) {};
  \node[base node] (A)    [right of=0]       {$a$};
  \node[base node] (notC) [right of=A]       {$\lnot c$};
  \node[base node] (B)    [right of=notC]    {$b$};
  \node[base node] (notD) [right of=B]       {$\lnot d$};
  \node[base node] (D)    [below of=notD]    {$d$};
  \node[base node] (bot)  [above right of=D] {$\bot$};
  \path[]
      (0)    edge [] node {$9$}    (A)
      (A)    edge [] node {$1$}    (notC)
      (notC) edge [] node {$2$}    (B)
      (B)    edge [] node {$3$}    (D)
      (B)    edge [] node {$7$}    (notD)
      (notD) edge [] node {}       (bot)
      (D)    edge [] node {}       (bot);
\end{conflictgraph}

\begin{prooftree}
\AxiomC{$7. \; \lnot b \lor c \lor d$}
\AxiomC{$3. \; \lnot b \lor \lnot d$}
	\BinaryInfC{$\lnot b \lor c$}
	\AxiomC{$2. \; b \lor c$}
	  \BinaryInfC{$c$}
		\AxiomC{$1. \; \lnot a \lor \lnot c$}
			\BinaryInfC{$\lnot a$}
		  \AxiomC{$9. \; a$}
			  \BinaryInfC{$\bot$}
\end{prooftree}