Questionnaire-2024/sat_solvers_dpll/1018_sol.tex

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

Conflict in step 4\\
\scalebox{0.75}{

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%%
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  % Edge: 1 -> 4
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  % Edge: 5 -> 2
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  % Node: 1
\begin{scope}
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  % Node: 4
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  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,31.0bp) node {$\lnot c$};
\end{scope}
  % Node: 2
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (242.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (242.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 3
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  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,51.0bp) node {$\lnot b$};
\end{scope}
  % Node: 5
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
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  \draw (183.0bp,11.0bp) node {$b$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$2. \; \lnot b \lor c$}
	\AxiomC{$4. \; b \lor c$}
	\BinaryInfC{$c$}
	\AxiomC{$3. \; \lnot a \lor \lnot c$}
	\BinaryInfC{$\lnot a$}
\end{prooftree}

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

Conflict in step 7\\
\scalebox{0.75}{

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%%
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  % Node: 4
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  \draw (76.0bp,31.0bp) node {$\lnot a$};
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  % Node: 1
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  % Node: 2
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  \draw (162.0bp,51.0bp) node {$b$};
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  % Node: 3
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  \draw (162.0bp,11.0bp) node {$\lnot b$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$1. \; a \lor b$}
	\AxiomC{$5. \; a \lor \lnot b$}
	\BinaryInfC{$a$}
	\AxiomC{$6. \; \lnot a$}
	\BinaryInfC{$\bot$}
\end{prooftree}