Questionnaire-2024/sat_solvers_dpll/0039_sol.tex

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

Conflict in step 5\\
\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 {$$1$$};
  % Edge: 3 -> 4
  \draw [->] (107.97bp,33.373bp) .. controls (121.23bp,36.53bp) and (144.98bp,42.186bp)  .. (172.05bp,48.63bp);
  \draw (140.0bp,50.5bp) node {$$2$$};
  % Edge: 3 -> 5
  \draw [->] (107.11bp,26.371bp) .. controls (112.48bp,23.828bp) and (119.48bp,20.828bp)  .. (126.0bp,19.0bp) .. controls (137.65bp,15.733bp) and (151.06bp,13.758bp)  .. (171.92bp,11.639bp);
  \draw (140.0bp,26.5bp) node {$$7$$};
  % Edge: 4 -> 2
  \draw [->] (193.67bp,47.438bp) .. controls (201.41bp,44.585bp) and (212.51bp,40.495bp)  .. (231.47bp,33.51bp);
  % Edge: 5 -> 2
  \draw [->] (193.67bp,14.223bp) .. controls (201.24bp,16.746bp) and (212.02bp,20.339bp)  .. (231.09bp,26.697bp);
  % 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 {$d$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (242.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (242.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,51.0bp) node {$e$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,11.0bp) node {$\lnot e$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$2. \; \lnot d \lor e$}
	\AxiomC{$7. \; \lnot d \lor \lnot e$}
	\BinaryInfC{$\lnot d$}
	\AxiomC{$1. \; b \lor d$}
	\BinaryInfC{$b$}
\end{prooftree}

\hspace{-0.09cm}\scalebox{0.85}{
	\begin{dplltabular}{5}
		\dpllStep{6|7|8|9|10}
		\dpllDecL{0|0|0|0|0}
		\dpllAssi{$a$|$a, b$|$a, b, c$|\makecell{$a, b, c, $ \\ $d$}|\makecell{$a, b, c, $ \\ $d, \lnot e$}}
		\dpllClause{1}{$b, d$}{$b, d$|\done|\done|\done|\done}
		\dpllClause{2}{$\lnot d, e$}{$\lnot d, e$|$\lnot d, e$|$\lnot d, e$|$e$|\done}
		\dpllClause{3}{$a, d$}{\done|\done|\done|\done|\done}
		\dpllClause{4}{$\lnot b, c$}{$\lnot b, c$|$c$|\done|\done|\done}
		\dpllClause{5}{$\lnot c, d$}{$\lnot c, d$|$\lnot c, d$|$d$|\done|\done}
		\dpllClause{6}{$\lnot c, e$}{$\lnot c, e$|$\lnot c, e$|$e$|$e$|\conflict}
		\dpllClause{7}{$\lnot d, \lnot e$}{$\lnot d, \lnot e$|$\lnot d, \lnot e$|$\lnot d, \lnot e$|$\lnot e$|\done}
		\dpllClause{8}{$b$}{$b$|\done|\done|\done|\done}
		\dpllBCP{$b$|$c$|$d$|$\lnot e$| - }
		\dpllPL{ - | - | - | - | - }
		\dpllDeci{ - | - | - | - |UNSAT}
	\end{dplltabular}
}

Conflict in step 10\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 0 -> 7
  \draw [->] (1.0536bp,30.0bp) .. controls (2.5071bp,30.0bp) and (33.188bp,30.0bp)  .. (64.696bp,30.0bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (33.0bp,37.5bp) node {$$8$$};
  % Edge: 2 -> 1
  \draw [->] (343.41bp,84.098bp) .. controls (351.75bp,78.249bp) and (364.75bp,69.126bp)  .. (383.55bp,55.926bp);
  % 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,49.5bp) node {$$5$$};
  % Edge: 3 -> 5
  \draw [->] (172.07bp,25.196bp) .. controls (177.42bp,22.594bp) and (184.42bp,19.594bp)  .. (191.0bp,18.0bp) .. controls (217.72bp,11.525bp) and (249.6bp,10.402bp)  .. (280.0bp,10.619bp);
  \draw (205.0bp,25.5bp) node {$$6$$};
  % Edge: 4 -> 2
  \draw [->] (257.63bp,55.72bp) .. controls (263.06bp,59.202bp) and (270.3bp,63.616bp)  .. (277.0bp,67.0bp) .. controls (288.94bp,73.032bp) and (302.86bp,78.753bp)  .. (323.28bp,86.537bp);
  \draw (291.0bp,86.5bp) node {$$2$$};
  % Edge: 4 -> 6
  \draw [->] (259.3bp,50.0bp) .. controls (272.47bp,50.0bp) and (295.58bp,50.0bp)  .. (322.88bp,50.0bp);
  \draw (291.0bp,57.5bp) node {$$7$$};
  % Edge: 5 -> 1
  \draw [->] (301.83bp,14.333bp) .. controls (312.52bp,18.001bp) and (330.07bp,24.154bp)  .. (345.0bp,30.0bp) .. controls (354.39bp,33.678bp) and (364.72bp,38.037bp)  .. (382.7bp,45.855bp);
  % Edge: 6 -> 1
  \draw [->] (345.19bp,50.0bp) .. controls (352.57bp,50.0bp) and (362.78bp,50.0bp)  .. (381.9bp,50.0bp);
  % Edge: 7 -> 3
  \draw [->] (87.3bp,30.0bp) .. controls (100.47bp,30.0bp) and (123.58bp,30.0bp)  .. (150.88bp,30.0bp);
  \draw (119.0bp,37.5bp) node {$$4$$};
  % Node: 7
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (76.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (76.0bp,30.0bp) node {$b$};
\end{scope}
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (393.0bp,50.0bp) ellipse (11.0bp and 11.0bp);
  \draw (393.0bp,50.0bp) node {$\bot$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (334.0bp,90.0bp) ellipse (11.0bp and 11.0bp);
  \draw (334.0bp,90.0bp) node {$e$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (162.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (162.0bp,30.0bp) node {$c$};
\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{$2. \; \lnot d \lor e$}
	\AxiomC{$7. \; \lnot d \lor \lnot e$}
	\BinaryInfC{$\lnot d$}
	\AxiomC{$5. \; \lnot c \lor d$}
	\BinaryInfC{$\lnot c$}
	\AxiomC{$6. \; \lnot c \lor e$}
	\BinaryInfC{$\lnot c \lor e$}
	\AxiomC{$4. \; \lnot b \lor c$}
	\BinaryInfC{$e \lor \lnot b$}
	\AxiomC{$8. \; b$}
	\BinaryInfC{$e$}
\end{prooftree}