Questionnaire-2024/sat_solvers_dpll/4004_sol.tex

\begin{dplltabular}{6}
  \dpllStep{1|2|3|4|5|6}
  \dpllDecL{0|1|2|2|2|2}

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


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

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

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

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

\begin{conflictgraph}
  \node[base node] (notA)                       {$\lnot a$};
  \node[base node] (notB) [below of=notA]       {$\lnot b$};
  \node[base node] (C)    [right of=notA]       {$c$};
  \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 {$1$} (C)
      (notB) edge [] node {$1$} (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{$1. \; a \lor b \lor c$}
	  	\BinaryInfC{$a \lor b$}
\end{prooftree}


\begin{dplltabular}{9}
  \dpllStep{(2)|7|8|9|10}
  \dpllDecL{1  |1|1|1|1}

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


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

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

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

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

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

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

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

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