Questionnaire-2024/sat_solvers_dpll/1024_sol.tex

\hspace{-0.09cm}\scalebox{0.85}{
	\begin{dplltabular}{5}
		\dpllStep{1|2|3|4|5}
		\dpllDecL{0|1|1|1|1}
		\dpllAssi{ - |$\lnot a$|$\lnot a, \lnot c$|$\lnot a, \lnot c, b$|\makecell{$\lnot a, \lnot c, b, $ \\ $\lnot d$}}
		\dpllClause{1}{$\lnot b, c, d$}{$\lnot b, c, d$|$\lnot b, c, d$|$\lnot b, d$|$d$|\conflict}
		\dpllClause{2}{$\lnot b, \lnot d$}{$\lnot b, \lnot d$|$\lnot b, \lnot d$|$\lnot b, \lnot d$|$\lnot d$|\done}
		\dpllClause{3}{$a, \lnot c$}{$a, \lnot c$|$\lnot c$|\done|\done|\done}
		\dpllClause{4}{$\lnot c, e$}{$\lnot c, e$|$\lnot c, e$|\done|\done|\done}
		\dpllClause{5}{$b, c$}{$b, c$|$b, c$|$b$|\done|\done}
		\dpllClause{6}{$\lnot a, \lnot e$}{$\lnot a, \lnot e$|\done|\done|\done|\done}
		\dpllBCP{ - |$\lnot c$|$b$|$\lnot d$| - }
		\dpllPL{ - | - | - | - | - }
		\dpllDeci{$\lnot a$| - | - | - | - }
	\end{dplltabular}
}

Conflict in step 5\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
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%%
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  % Edge: 1 -> 5
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  % Edge: 5 -> 6
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  % Edge: 6 -> 4
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  \draw (232.0bp,28.5bp) node {$$2$$};
  % Node: 1
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,56.0bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,56.0bp) node {$\lnot a$};
\end{scope}
  % Node: 5
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,56.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,56.0bp) node {$\lnot c$};
\end{scope}
  % Node: 2
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (334.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (334.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 3
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (275.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (275.0bp,51.0bp) node {$d$};
\end{scope}
  % Node: 4
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (275.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (275.0bp,11.0bp) node {$\lnot d$};
\end{scope}
  % Node: 6
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (186.0bp,32.0bp) ellipse (11.0bp and 11.0bp);
  \draw (186.0bp,32.0bp) node {$b$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$1. \; \lnot b \lor c \lor d$}
	\AxiomC{$2. \; \lnot b \lor \lnot d$}
	\BinaryInfC{$\lnot b \lor c$}
	\AxiomC{$5. \; b \lor c$}
	\BinaryInfC{$c$}
	\AxiomC{$3. \; a \lor \lnot c$}
	\BinaryInfC{$a$}
\end{prooftree}

\hspace{-0.09cm}\scalebox{0.85}{
	\begin{dplltabular}{6}
		\dpllStep{6|7|8|9|10|11}
		\dpllDecL{0|0|0|0|0|0}
		\dpllAssi{ - |$a$|$a, \lnot e$|$a, \lnot e, \lnot c$|\makecell{$a, \lnot e, \lnot c, $ \\ $b$}|\makecell{$a, \lnot e, \lnot c, $ \\ $b, \lnot d$}}
		\dpllClause{1}{$\lnot b, c, d$}{$\lnot b, c, d$|$\lnot b, c, d$|$\lnot b, c, d$|$\lnot b, d$|$d$|\conflict}
		\dpllClause{2}{$\lnot b, \lnot d$}{$\lnot b, \lnot d$|$\lnot b, \lnot d$|$\lnot b, \lnot d$|$\lnot b, \lnot d$|$\lnot d$|\done}
		\dpllClause{3}{$a, \lnot c$}{$a, \lnot c$|\done|\done|\done|\done|\done}
		\dpllClause{4}{$\lnot c, e$}{$\lnot c, e$|$\lnot c, e$|$\lnot c$|\done|\done|\done}
		\dpllClause{5}{$b, c$}{$b, c$|$b, c$|$b, c$|$b$|\done|\done}
		\dpllClause{6}{$\lnot a, \lnot e$}{$\lnot a, \lnot e$|$\lnot e$|\done|\done|\done|\done}
		\dpllClause{7}{$a$}{$a$|\done|\done|\done|\done|\done}
		\dpllBCP{$a$|$\lnot e$|$\lnot c$|$b$|$\lnot d$| - }
		\dpllPL{ - | - | - | - | - | - }
		\dpllDeci{ - | - | - | - | - |UNSAT}
	\end{dplltabular}
}

Conflict in step 11\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
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  % Edge: 0 -> 7
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  \pgfsetstrokecolor{strokecol}
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  % Edge: 2 -> 1
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  % Edge: 3 -> 1
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  % Edge: 4 -> 2
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  % Edge: 4 -> 5
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  % Edge: 5 -> 2
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  % Edge: 5 -> 3
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  % Edge: 6 -> 4
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  % Edge: 7 -> 6
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  % Node: 7
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  % Node: 1
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  \draw (485.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 2
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  \draw (426.0bp,51.0bp) node {$d$};
\end{scope}
  % Node: 3
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (426.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (426.0bp,11.0bp) node {$\lnot d$};
\end{scope}
  % Node: 4
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
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  \draw (248.0bp,56.0bp) node {$\lnot c$};
\end{scope}
  % Node: 5
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
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  \draw (337.0bp,32.0bp) node {$b$};
\end{scope}
  % Node: 6
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (162.0bp,56.0bp) ellipse (11.0bp and 11.0bp);
  \draw (162.0bp,56.0bp) node {$\lnot e$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$1. \; \lnot b \lor c \lor d$}
	\AxiomC{$2. \; \lnot b \lor \lnot d$}
	\BinaryInfC{$\lnot b \lor c$}
	\AxiomC{$5. \; b \lor c$}
	\BinaryInfC{$c$}
	\AxiomC{$4. \; \lnot c \lor e$}
	\BinaryInfC{$e$}
	\AxiomC{$6. \; \lnot a \lor \lnot e$}
	\BinaryInfC{$\lnot a$}
	\AxiomC{$7. \; a$}
	\BinaryInfC{$\bot$}
\end{prooftree}