Questionnaire-2024/sat_solvers_dpll/0051_sol.tex

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

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

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 1 -> 3
  \draw [->] (22.3bp,31.0bp) .. controls (35.47bp,31.0bp) and (58.583bp,31.0bp)  .. (85.878bp,31.0bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (54.0bp,38.5bp) node {$$3$$};
  % Edge: 3 -> 6
  \draw [->] (108.3bp,31.0bp) .. controls (121.47bp,31.0bp) and (144.58bp,31.0bp)  .. (171.88bp,31.0bp);
  \draw (140.0bp,38.5bp) node {$$7$$};
  % Edge: 4 -> 2
  \draw [->] (279.67bp,47.438bp) .. controls (287.41bp,44.585bp) and (298.51bp,40.495bp)  .. (317.47bp,33.51bp);
  % Edge: 5 -> 2
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  % Edge: 6 -> 4
  \draw [->] (193.97bp,33.373bp) .. controls (207.23bp,36.53bp) and (230.98bp,42.186bp)  .. (258.05bp,48.63bp);
  \draw (226.0bp,50.5bp) node {$$5$$};
  % Edge: 6 -> 5
  \draw [->] (193.11bp,26.371bp) .. controls (198.48bp,23.828bp) and (205.48bp,20.828bp)  .. (212.0bp,19.0bp) .. controls (223.65bp,15.733bp) and (237.06bp,13.758bp)  .. (257.92bp,11.639bp);
  \draw (226.0bp,26.5bp) node {$$6$$};
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,31.0bp) node {$\lnot b$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,31.0bp) node {$c$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (328.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (328.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,31.0bp) node {$\lnot e$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (269.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (269.0bp,51.0bp) node {$\lnot d$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (269.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (269.0bp,11.0bp) node {$d$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$5. \; \lnot d \lor e$}
	\AxiomC{$6. \; d \lor e$}
	\BinaryInfC{$e$}
	\AxiomC{$7. \; \lnot c \lor \lnot e$}
	\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{0|0|0|0|0}
		\dpllAssi{$\lnot a$|$\lnot a, b$|$\lnot a, b, \lnot d$|\makecell{$\lnot a, b, \lnot d, $ \\ $c$}|\makecell{$\lnot a, b, \lnot d, $ \\ $c, \lnot e$}}
		\dpllClause{1}{$b, c, d$}{$b, c, d$|\done|\done|\done|\done}
		\dpllClause{2}{$\lnot a, \lnot c$}{\done|\done|\done|\done|\done}
		\dpllClause{3}{$b, c$}{$b, c$|\done|\done|\done|\done}
		\dpllClause{4}{$\lnot b, \lnot d$}{$\lnot b, \lnot d$|$\lnot d$|\done|\done|\done}
		\dpllClause{5}{$\lnot d, e$}{$\lnot d, e$|$\lnot d, e$|\done|\done|\done}
		\dpllClause{6}{$d, e$}{$d, e$|$d, e$|$e$|$e$|\conflict}
		\dpllClause{7}{$\lnot c, \lnot e$}{$\lnot c, \lnot e$|$\lnot c, \lnot e$|$\lnot c, \lnot e$|$\lnot e$|\done}
		\dpllClause{8}{$\lnot b, c, d$}{$\lnot b, c, d$|$c, d$|$c$|\done|\done}
		\dpllClause{9}{$b$}{$b$|\done|\done|\done|\done}
		\dpllBCP{$b$|$\lnot d$|$c$|$\lnot e$| - }
		\dpllPL{ - | - | - | - | - }
		\dpllDeci{ - | - | - | - |UNSAT}
	\end{dplltabular}
}

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

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 0 -> 6
  \draw [->] (1.0536bp,11.0bp) .. controls (2.5071bp,11.0bp) and (33.188bp,11.0bp)  .. (64.696bp,11.0bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (33.0bp,18.5bp) node {$$9$$};
  % Edge: 2 -> 3
  \draw [->] (176.2bp,41.119bp) .. controls (197.65bp,45.574bp) and (247.78bp,55.985bp)  .. (285.85bp,63.891bp);
  \draw (211.0bp,57.5bp) node {$$6$$};
  % Edge: 2 -> 5
  \draw [->] (175.6bp,35.143bp) .. controls (181.64bp,32.91bp) and (189.65bp,30.291bp)  .. (197.0bp,29.0bp) .. controls (208.81bp,26.927bp) and (222.22bp,26.386bp)  .. (243.0bp,26.532bp);
  \draw (211.0bp,36.5bp) node {$$8$$};
  % Edge: 3 -> 1
  \draw [->] (307.95bp,64.009bp) .. controls (324.28bp,60.745bp) and (357.03bp,54.194bp)  .. (388.09bp,47.983bp);
  % Edge: 4 -> 1
  \draw [->] (350.67bp,30.223bp) .. controls (358.24bp,32.746bp) and (369.02bp,36.339bp)  .. (388.09bp,42.697bp);
  % Edge: 5 -> 4
  \draw [->] (265.3bp,27.0bp) .. controls (278.47bp,27.0bp) and (301.58bp,27.0bp)  .. (328.88bp,27.0bp);
  \draw (297.0bp,34.5bp) node {$$7$$};
  % Edge: 6 -> 2
  \draw [->] (86.643bp,14.103bp) .. controls (100.56bp,18.582bp) and (126.57bp,26.952bp)  .. (154.31bp,35.883bp);
  \draw (119.0bp,35.5bp) node {$$4$$};
  % Edge: 6 -> 5
  \draw [->] (86.905bp,8.8911bp) .. controls (105.2bp,5.3124bp) and (145.48bp,-1.0889bp)  .. (179.0bp,4.0bp) .. controls (198.27bp,6.925bp) and (219.42bp,13.903bp)  .. (243.64bp,23.083bp);
  \draw (165.0bp,11.5bp) node {$$8$$};
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (76.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (76.0bp,11.0bp) node {$b$};
\end{scope}
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (399.0bp,46.0bp) ellipse (11.0bp and 11.0bp);
  \draw (399.0bp,46.0bp) node {$\bot$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (165.0bp,39.0bp) ellipse (11.0bp and 11.0bp);
  \draw (165.0bp,39.0bp) node {$\lnot d$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (297.0bp,66.0bp) ellipse (11.0bp and 11.0bp);
  \draw (297.0bp,66.0bp) node {$e$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (254.0bp,27.0bp) ellipse (11.0bp and 11.0bp);
  \draw (254.0bp,27.0bp) node {$c$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (340.0bp,27.0bp) ellipse (11.0bp and 11.0bp);
  \draw (340.0bp,27.0bp) node {$\lnot e$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$7. \; \lnot c \lor \lnot e$}
	\AxiomC{$6. \; d \lor e$}
	\BinaryInfC{$\lnot c \lor d$}
	\AxiomC{$8. \; \lnot b \lor c \lor d$}
	\BinaryInfC{$d \lor \lnot b$}
	\AxiomC{$4. \; \lnot b \lor \lnot d$}
	\BinaryInfC{$\lnot b$}
	\AxiomC{$9. \; b$}
	\BinaryInfC{$\bot$}
\end{prooftree}