Questionnaire-2024/sat_solvers_dpll/0037_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}{$a, \lnot b, c$}{$a, \lnot b, c$|$\lnot b, c$|\done|\done|\done|\done}
		\dpllClause{2}{$\lnot a, b$}{$\lnot a, b$|\done|\done|\done|\done|\done}
		\dpllClause{3}{$b, c$}{$b, c$|$b, c$|$c$|\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,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 1 -> 4
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  \pgfsetstrokecolor{strokecol}
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  % Edge: 4 -> 5
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  \draw (140.0bp,50.0bp) node {$$4$$};
  % Edge: 4 -> 6
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  \draw (140.0bp,26.0bp) node {$$5$$};
  % Edge: 5 -> 7
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  \draw (226.0bp,58.0bp) node {$$6$$};
  % Edge: 6 -> 3
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  % Edge: 7 -> 3
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  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,30.5bp) node {$\lnot b$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,30.5bp) node {$c$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,70.5bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,70.5bp) node {$\lnot a$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (328.0bp,23.5bp) ellipse (11.0bp and 11.0bp);
  \draw (328.0bp,23.5bp) node {$\bot$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,50.5bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,50.5bp) node {$d$};
\end{scope}
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (226.0bp,11.5bp) ellipse (11.0bp and 11.0bp);
  \draw (226.0bp,11.5bp) node {$e$};
\end{scope}
  % Node: 7
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (269.0bp,50.5bp) ellipse (11.0bp and 11.0bp);
  \draw (269.0bp,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{$3. \; b \lor c$}
	\BinaryInfC{$b$}
\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}{$a, \lnot b, c$}{$\lnot b, c$|$c$|\done|\done|\done}
		\dpllClause{2}{$\lnot a, b$}{\done|\done|\done|\done|\done}
		\dpllClause{3}{$b, c$}{$b, c$|\done|\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}{$b$}{$b$|\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: 0 -> 7
  \draw [->] (1.0536bp,51.0bp) .. controls (2.5071bp,51.0bp) and (33.188bp,51.0bp)  .. (64.696bp,51.0bp);
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  \pgfsetstrokecolor{strokecol}
  \draw (33.0bp,58.5bp) node {$$7$$};
  % Edge: 1 -> 3
  \draw [->] (87.161bp,9.8516bp) .. controls (98.575bp,8.9047bp) and (117.53bp,8.3716bp)  .. (133.0bp,13.0bp) .. controls (136.82bp,14.143bp) and (140.65bp,15.935bp)  .. (152.72bp,23.413bp);
  \draw (119.0bp,20.5bp) node {$$1$$};
  % Edge: 3 -> 4
  \draw [->] (172.97bp,32.373bp) .. controls (186.23bp,35.53bp) and (209.98bp,41.186bp)  .. (237.05bp,47.63bp);
  \draw (205.0bp,50.5bp) node {$$4$$};
  % Edge: 3 -> 5
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  \draw (205.0bp,26.5bp) node {$$5$$};
  % Edge: 4 -> 6
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  % Edge: 5 -> 2
  \draw [->] (301.95bp,12.194bp) .. controls (318.12bp,14.134bp) and (350.4bp,18.008bp)  .. (381.79bp,21.775bp);
  % Edge: 6 -> 2
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  % Edge: 7 -> 3
  \draw [->] (86.966bp,48.509bp) .. controls (100.23bp,45.193bp) and (123.98bp,39.255bp)  .. (151.05bp,32.488bp);
  \draw (119.0bp,51.5bp) node {$$1$$};
  % Node: 7
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (76.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (76.0bp,51.0bp) node {$b$};
\end{scope}
  % Node: 1
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (76.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (76.0bp,11.0bp) node {$\lnot a$};
\end{scope}
  % Node: 3
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
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  \draw (162.0bp,30.0bp) node {$c$};
\end{scope}
  % Node: 2
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (393.0bp,23.0bp) ellipse (11.0bp and 11.0bp);
  \draw (393.0bp,23.0bp) node {$\bot$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (248.0bp,50.0bp) ellipse (11.0bp and 11.0bp);
  \draw (248.0bp,50.0bp) node {$d$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (291.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (291.0bp,11.0bp) node {$e$};
\end{scope}
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (334.0bp,50.0bp) ellipse (11.0bp and 11.0bp);
  \draw (334.0bp,50.0bp) 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{$1. \; a \lor \lnot b \lor c$}
	\BinaryInfC{$a \lor \lnot b$}
	\AxiomC{$7. \; b$}
	\BinaryInfC{$a$}
\end{prooftree}