Questionnaire-2024/sat_solvers_dpll/ex2_sol.tex

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

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

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%%
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  % Edge: 1 -> 4
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  \draw (11.0bp,88.5bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,88.5bp) node {$\lnot b$};
\end{scope}
  % Node: 4
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  \pgfsetstrokecolor{strokecol}
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  \draw (97.0bp,67.5bp) node {$c$};
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  \pgfsetstrokecolor{strokecol}
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  \draw (140.0bp,106.5bp) node {$\lnot d$};
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  % Node: 2
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  \draw (183.0bp,67.5bp) node {$d$};
\end{scope}
%
\end{tikzpicture}


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

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