Questionnaire-2024/sat_solvers_dpll/1026_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{ - |$\lnot a$|$\lnot a, \lnot b$|$\lnot a, \lnot b, e$|\makecell{$\lnot a, \lnot b, e, $ \\ $\lnot c$}|\makecell{$\lnot a, \lnot b, e, $ \\ $\lnot c, \lnot d$}}
		\dpllClause{1}{$\lnot b, d, e$}{$\lnot b, d, e$|$\lnot b, d, e$|\done|\done|\done|\done}
		\dpllClause{2}{$b, e$}{$b, e$|$b, e$|$e$|\done|\done|\done}
		\dpllClause{3}{$c, d$}{$c, d$|$c, d$|$c, d$|$c, d$|$d$|\conflict}
		\dpllClause{4}{$\lnot a, \lnot e$}{$\lnot a, \lnot e$|\done|\done|\done|\done|\done}
		\dpllClause{5}{$a, \lnot c, \lnot e$}{$a, \lnot c, \lnot e$|$\lnot c, \lnot e$|$\lnot c, \lnot e$|$\lnot c$|\done|\done}
		\dpllClause{6}{$c, \lnot d$}{$c, \lnot d$|$c, \lnot d$|$c, \lnot d$|$c, \lnot d$|$\lnot d$|\done}
		\dpllClause{7}{$\lnot b, \lnot d$}{$\lnot b, \lnot d$|$\lnot b, \lnot d$|\done|\done|\done|\done}
		\dpllBCP{ - | - |$e$|$\lnot c$|$\lnot d$| - }
		\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
  \draw [->] (22.3bp,52.0bp) .. controls (35.47bp,52.0bp) and (58.583bp,52.0bp)  .. (85.878bp,52.0bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (54.0bp,59.5bp) node {$$2$$};
  % Edge: 2 -> 6
  \draw [->] (108.16bp,10.852bp) .. controls (119.58bp,9.9047bp) and (138.53bp,9.3716bp)  .. (154.0bp,14.0bp) .. controls (157.82bp,15.143bp) and (161.65bp,16.935bp)  .. (173.72bp,24.413bp);
  \draw (140.0bp,21.5bp) node {$$5$$};
  % Edge: 4 -> 6
  \draw [->] (107.97bp,49.509bp) .. controls (121.23bp,46.193bp) and (144.98bp,40.255bp)  .. (172.05bp,33.488bp);
  \draw (140.0bp,51.5bp) node {$$5$$};
  % Edge: 5 -> 3
  \draw [->] (279.67bp,47.438bp) .. controls (287.41bp,44.585bp) and (298.51bp,40.495bp)  .. (317.47bp,33.51bp);
  % Edge: 6 -> 5
  \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 {$$3$$};
  % Edge: 6 -> 7
  \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 {$$6$$};
  % Edge: 7 -> 3
  \draw [->] (279.67bp,14.223bp) .. controls (287.24bp,16.746bp) and (298.02bp,20.339bp)  .. (317.09bp,26.697bp);
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,52.0bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,52.0bp) node {$\lnot b$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,52.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,52.0bp) node {$e$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,12.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,12.0bp) node {$\lnot a$};
\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 {$\lnot c$};
\end{scope}
  % Node: 3
\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: 5
\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 {$d$};
\end{scope}
  % Node: 7
\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 {$\lnot d$};
\end{scope}
%
\end{tikzpicture}


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

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

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 1 -> 5
  \draw [->] (21.668bp,76.625bp) .. controls (37.13bp,82.146bp) and (68.489bp,91.962bp)  .. (96.0bp,91.962bp) .. controls (96.0bp,91.962bp) and (96.0bp,91.962bp)  .. (276.0bp,91.962bp) .. controls (301.27bp,91.962bp) and (327.17bp,77.264bp)  .. (351.77bp,59.755bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (186.0bp,99.462bp) node {$$5$$};
  % Edge: 1 -> 7
  \draw [->] (21.31bp,69.082bp) .. controls (34.682bp,63.511bp) and (59.622bp,53.119bp)  .. (86.648bp,41.859bp);
  \draw (54.0bp,68.462bp) node {$$8$$};
  % Edge: 3 -> 2
  \draw [->] (286.28bp,13.897bp) .. controls (309.78bp,16.033bp) and (367.77bp,21.304bp)  .. (408.81bp,25.035bp);
  % Edge: 4 -> 5
  \draw [->] (286.3bp,52.962bp) .. controls (299.47bp,52.962bp) and (322.58bp,52.962bp)  .. (349.88bp,52.962bp);
  \draw (318.0bp,60.462bp) node {$$5$$};
  % Edge: 5 -> 2
  \draw [->] (371.16bp,48.624bp) .. controls (379.01bp,44.906bp) and (390.56bp,39.434bp)  .. (409.68bp,30.379bp);
  % Edge: 6 -> 3
  \draw [->] (196.33bp,33.514bp) .. controls (202.39bp,30.755bp) and (210.54bp,27.303bp)  .. (218.0bp,24.962bp) .. controls (229.69bp,21.294bp) and (243.1bp,18.377bp)  .. (263.94bp,14.536bp);
  \draw (232.0bp,32.462bp) node {$$1$$};
  % Edge: 6 -> 4
  \draw [->] (196.98bp,39.683bp) .. controls (210.8bp,42.066bp) and (236.09bp,46.425bp)  .. (263.87bp,51.215bp);
  \draw (232.0bp,56.462bp) node {$$3$$};
  % Edge: 7 -> 3
  \draw [->] (106.65bp,32.3bp) .. controls (120.01bp,23.928bp) and (146.76bp,8.629bp)  .. (172.0bp,2.9618bp) .. controls (200.25bp,-3.3831bp) and (233.81bp,2.3811bp)  .. (264.31bp,9.9958bp);
  \draw (186.0bp,10.462bp) node {$$1$$};
  % Edge: 7 -> 6
  \draw [->] (108.32bp,37.962bp) .. controls (122.2bp,37.962bp) and (147.16bp,37.962bp)  .. (174.96bp,37.962bp);
  \draw (140.0bp,45.462bp) node {$$7$$};
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (11.0bp,72.96bp) ellipse (11.0bp and 11.0bp);
  \draw (11.0bp,72.962bp) node {$\lnot a$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (361.0bp,52.96bp) ellipse (11.0bp and 11.0bp);
  \draw (361.0bp,52.962bp) node {$\lnot e$};
\end{scope}
  % Node: 7
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,37.96bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,37.962bp) node {$b$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (420.0bp,25.96bp) ellipse (11.0bp and 11.0bp);
  \draw (420.0bp,25.962bp) node {$\bot$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (275.0bp,12.96bp) ellipse (11.0bp and 11.0bp);
  \draw (275.0bp,12.962bp) node {$e$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (275.0bp,52.96bp) ellipse (11.0bp and 11.0bp);
  \draw (275.0bp,52.962bp) node {$c$};
\end{scope}
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (186.0bp,37.96bp) ellipse (11.0bp and 11.0bp);
  \draw (186.0bp,37.962bp) node {$\lnot d$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$5. \; a \lor \lnot c \lor \lnot e$}
	\AxiomC{$1. \; \lnot b \lor d \lor e$}
	\BinaryInfC{$a \lor \lnot c \lor \lnot b \lor d$}
	\AxiomC{$3. \; c \lor d$}
	\BinaryInfC{$a \lor \lnot b \lor d$}
	\AxiomC{$7. \; \lnot b \lor \lnot d$}
	\BinaryInfC{$a \lor \lnot b$}
	\AxiomC{$8. \; a \lor b$}
	\BinaryInfC{$a$}
\end{prooftree}