Questionnaire-2024/sat_solvers_dpll/4004_sol_second_part.tex

\begin{conflictgraph}
  \node[base node] (notA)                       {$\lnot a$};
  \node[base node] (B)    [right of=notA]       {$b$};
  \node[base node] (C)    [right of=B]          {$c$};
  \node[base node] (D)    [below right of=C]    {$d$};
  \node[base node] (E)    [right of=C]          {$e$};
  \node[base node] (notE) [right of=D]          {$\lnot e$};
  \node[base node] (bot)  [above right of=notE] {$\bot$};
  \path[]
      (notA) edge [] node {$7$} (B)
      (B)    edge [] node {$3$} (C)
      (C)    edge [] node {$4$} (D)
      (C)    edge [] node {$5$} (E)
      (D)    edge [] node {$6$} (notE)
      (notE) edge [] node {}    (bot)
      (E)    edge [] node {}    (bot);
\end{conflictgraph}

\begin{prooftree}
\AxiomC{$6. \; \lnot d \lor \lnot e$}
\AxiomC{$4. \; \lnot c \lor d$}
	\BinaryInfC{$\lnot e \lor \lnot c$}
	\AxiomC{$5. \; \lnot c \lor e$}
		\BinaryInfC{$\lnot c$}
	  \AxiomC{$3. \; \lnot b \lor c$}
	  	\BinaryInfC{$\lnot b$}
	    \AxiomC{$7. \; a \lor b$}
	    	\BinaryInfC{$a$}
\end{prooftree}

\begin{dplltabular}{9}
  \dpllStep{(1)|11|12|13|14|15}
  \dpllDecL{0  |0 |0 |0 |0 |0}

  \dpllAssi{-|
            $a$|
            $a, b$|
            $a, b, c$|
            $a, b, c, d$|
            $a, b, c, d, \lnot e$}


  \dpllClause{1}{$a,b,c$}
             {$a,b,c$|\done|\done|\done|\done|\done}

  \dpllClause{2}{$\lnot a,b$}
             {$\lnot a, b$|$b$|\done|\done|\done|\done}

  \dpllClause{3}{$\lnot b, c$}
             {$\lnot b, c$|$\lnot b,c$|$c$|\done|\done|\done}

  \dpllClause{4}{$\lnot c,d$}
             {$\lnot c,d$|$\lnot c,d$|$\lnot c,d$|$d$|\done|\done}

  \dpllClause{5}{$\lnot c,e$}
             {$\lnot c,e$|$\lnot c,e$|$\lnot c,e$|$e$|$e$|\conflict}

  \dpllClause{6}{$\lnot d, \lnot e$}
             {$\lnot d, \lnot e$|$\lnot d, \lnot e$|$\lnot d, \lnot e$|$\lnot e$|$\lnot e$|\done}

  \dpllClause{7}{$a, b$}
             {$a,b$|\done|\done|\done|\done|\done}

  \dpllClause{8}{$a$}
             {$a$|\done|\done|\done|\done|\done}

  \dpllBCP {$a$|$b$|$c$|$d$|$\lnot e$|-}
  \dpllPL  {-|-|-|-|-|-}
  \dpllDeci{-|-|-|-|-|UNSAT}
\end{dplltabular}

\begin{conflictgraph}
  \node (0){};
  \node[base node] (A)    [right of=0]          {$a$};
  \node[base node] (B)    [right of=A]          {$b$};
  \node[base node] (C)    [right of=B]          {$c$};
  \node[base node] (D)    [below right of=C]    {$d$};
  \node[base node] (E)    [right of=C]          {$e$};
  \node[base node] (notE) [right of=D]          {$\lnot e$};
  \node[base node] (bot)  [above right of=notE] {$\bot$};
  \path[]
      (0)    edge [] node {8}    (A)
      (A)    edge [] node {2} (B)
      (B)    edge [] node {3} (C)
      (C)    edge [] node {4} (D)
      (C)    edge [] node {5} (E)
      (D)    edge [] node {6} (notE)
      (notE) edge [] node {}    (bot)
      (E)    edge [] node {}    (bot);
\end{conflictgraph}

\begin{prooftree}
\AxiomC{$6. \; \lnot d \lor \lnot e$}
\AxiomC{$4. \; \lnot c \lor d$}
	\BinaryInfC{$\lnot e \lor \lnot c$}
	\AxiomC{$5. \; \lnot c \lor e$}
		\BinaryInfC{$\lnot c$}
	  \AxiomC{$3. \; \lnot b \lor c$}
	  	\BinaryInfC{$\lnot b$}
	    \AxiomC{$2. \; \lnot a \lor b$}
	    	\BinaryInfC{$\lnot a$}
	      \AxiomC{$8. \; a$}
          \BinaryInfC{$\bot$}
\end{prooftree}