Questionnaire-2024/sat_solvers_dpll/1027_sol.tex

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

Conflict in step 6\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
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%%
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  % Edge: 1 -> 4
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  % Edge: 2 -> 4
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  % Edge: 4 -> 5
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  % Edge: 4 -> 6
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  % Edge: 5 -> 7
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  \draw (226.0bp,78.0bp) node {$$6$$};
  % Edge: 6 -> 3
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  % Edge: 7 -> 3
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  % Node: 1
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,70.5bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,70.5bp) node {$\lnot b$};
\end{scope}
  % Node: 4
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,50.5bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,50.5bp) node {$c$};
\end{scope}
  % Node: 2
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,30.5bp) node {$\lnot a$};
\end{scope}
  % Node: 3
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (328.0bp,43.5bp) ellipse (11.0bp and 11.0bp);
  \draw (328.0bp,43.5bp) node {$\bot$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,70.5bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,70.5bp) node {$d$};
\end{scope}
  % Node: 6
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (226.0bp,31.5bp) ellipse (11.0bp and 11.0bp);
  \draw (226.0bp,31.5bp) node {$e$};
\end{scope}
  % Node: 7
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (269.0bp,70.5bp) ellipse (11.0bp and 11.0bp);
  \draw (269.0bp,70.5bp) node {$\lnot e$};
\end{scope}
%
\end{tikzpicture}


}
\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}

\hspace{-0.09cm}\scalebox{0.85}{
	\begin{dplltabular}{5}
		\dpllStep{7|8|9|10|11}
		\dpllDecL{1|1|1|1|1}
		\dpllAssi{$\lnot a$|$\lnot a, b$|$\lnot a, b, c$|\makecell{$\lnot a, b, c, $ \\ $d$}|\makecell{$\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}
}

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

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 1 -> 7
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  \pgfsetstrokecolor{strokecol}
  \draw (54.0bp,38.0bp) node {$$7$$};
  % Edge: 3 -> 4
  \draw [->] (193.97bp,32.873bp) .. controls (207.23bp,36.03bp) and (230.98bp,41.686bp)  .. (258.05bp,48.13bp);
  \draw (226.0bp,50.0bp) node {$$4$$};
  % Edge: 3 -> 5
  \draw [->] (193.07bp,25.696bp) .. controls (198.42bp,23.094bp) and (205.42bp,20.094bp)  .. (212.0bp,18.5bp) .. controls (238.72bp,12.025bp) and (270.6bp,10.902bp)  .. (301.0bp,11.119bp);
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  % Edge: 4 -> 6
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  % Edge: 5 -> 2
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  % Edge: 6 -> 2
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  % Edge: 7 -> 3
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  \draw (140.0bp,38.0bp) node {$$3$$};
  % Node: 1
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  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,30.5bp) node {$\lnot a$};
\end{scope}
  % Node: 7
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,30.5bp) node {$b$};
\end{scope}
  % Node: 2
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\end{scope}
  % Node: 3
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  \pgfsetstrokecolor{strokecol}
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  \draw (183.0bp,30.5bp) node {$c$};
\end{scope}
  % Node: 4
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (269.0bp,50.5bp) ellipse (11.0bp and 11.0bp);
  \draw (269.0bp,50.5bp) node {$d$};
\end{scope}
  % Node: 5
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (312.0bp,11.5bp) ellipse (11.0bp and 11.0bp);
  \draw (312.0bp,11.5bp) node {$e$};
\end{scope}
  % Node: 6
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (355.0bp,50.5bp) ellipse (11.0bp and 11.0bp);
  \draw (355.0bp,50.5bp) node {$\lnot e$};
\end{scope}
%
\end{tikzpicture}


}
\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}