Questionnaire-2024/sat_solvers_dpll/0023_sol.tex

\hspace{-0.09cm}\scalebox{0.85}{
	\begin{dplltabular}{6}
		\dpllStep{1|2|3|4|5|6}
		\dpllDecL{0|1|1|1|1|1}
		\dpllAssi{ - |$\lnot u$|$\lnot u, v$|$\lnot u, v, \lnot y$|\makecell{$\lnot u, v, \lnot y, $ \\ $w$}|\makecell{$\lnot u, v, \lnot y, $ \\ $w, \lnot z$}}
		\dpllClause{1}{$u, v$}{$u, v$|$v$|\done|\done|\done|\done}
		\dpllClause{2}{$\lnot u, w$}{$\lnot u, w$|\done|\done|\done|\done|\done}
		\dpllClause{3}{$v, \lnot x$}{$v, \lnot x$|$v, \lnot x$|\done|\done|\done|\done}
		\dpllClause{4}{$w, y$}{$w, y$|$w, y$|$w, y$|$w$|\done|\done}
		\dpllClause{5}{$\lnot v, \lnot y$}{$\lnot v, \lnot y$|$\lnot v, \lnot y$|$\lnot y$|\done|\done|\done}
		\dpllClause{6}{$\lnot w, z$}{$\lnot w, z$|$\lnot w, z$|$\lnot w, z$|$\lnot w, z$|$z$|\conflict}
		\dpllClause{7}{$\lnot z, x$}{$\lnot z, x$|$\lnot z, x$|$\lnot z, x$|$\lnot z, x$|$\lnot z, x$|\done}
		\dpllClause{8}{$y, \lnot z$}{$y, \lnot z$|$y, \lnot z$|$y, \lnot z$|$\lnot z$|$\lnot z$|\done}
		\dpllClause{9}{$x, \lnot u$}{$x, \lnot u$|\done|\done|\done|\done|\done}
		\dpllBCP{ - |$v$|$\lnot y$|$w$|$\lnot z$| - }
		\dpllPL{ - | - | - | - | - | - }
		\dpllDeci{$\lnot u$| - | - | - | - | - }
	\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,30.5bp) .. controls (35.47bp,30.5bp) and (58.583bp,30.5bp)  .. (85.878bp,30.5bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (54.0bp,38.0bp) node {$$1$$};
  % Edge: 3 -> 5
  \draw [->] (108.3bp,30.5bp) .. controls (121.47bp,30.5bp) and (144.58bp,30.5bp)  .. (171.88bp,30.5bp);
  \draw (140.0bp,38.0bp) node {$$5$$};
  % Edge: 4 -> 6
  \draw [->] (280.3bp,50.5bp) .. controls (293.47bp,50.5bp) and (316.58bp,50.5bp)  .. (343.88bp,50.5bp);
  \draw (312.0bp,58.0bp) node {$$6$$};
  % Edge: 5 -> 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: 5 -> 7
  \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);
  \draw (226.0bp,26.0bp) node {$$8$$};
  % Edge: 6 -> 2
  \draw [->] (365.16bp,46.162bp) .. controls (373.01bp,42.445bp) and (384.56bp,36.972bp)  .. (403.68bp,27.917bp);
  % Edge: 7 -> 2
  \draw [->] (322.95bp,12.694bp) .. controls (339.12bp,14.634bp) and (371.4bp,18.508bp)  .. (402.79bp,22.275bp);
  % 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 {$\lnot u$};
\end{scope}
  % Node: 3
\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 {$v$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (414.0bp,23.5bp) ellipse (11.0bp and 11.0bp);
  \draw (414.0bp,23.5bp) node {$\bot$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (183.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  \draw (183.0bp,30.5bp) node {$\lnot y$};
\end{scope}
  % Node: 4
\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 {$w$};
\end{scope}
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (355.0bp,50.5bp) ellipse (11.0bp and 11.0bp);
  \draw (355.0bp,50.5bp) node {$z$};
\end{scope}
  % Node: 7
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (312.0bp,11.5bp) ellipse (11.0bp and 11.0bp);
  \draw (312.0bp,11.5bp) node {$\lnot z$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$6. \; \lnot w \lor z$}
	\AxiomC{$4. \; w \lor y$}
	\BinaryInfC{$z \lor y$}
	\AxiomC{$8. \; y \lor \lnot z$}
	\BinaryInfC{$y$}
	\AxiomC{$5. \; \lnot v \lor \lnot y$}
	\BinaryInfC{$\lnot v$}
	\AxiomC{$1. \; u \lor v$}
	\BinaryInfC{$u$}
\end{prooftree}

\hspace{-0.09cm}\scalebox{0.85}{
	\begin{dplltabular}{7}
		\dpllStep{7|8|9|10|11|12|13}
		\dpllDecL{0|0|0|0|0|0|0}
		\dpllAssi{ - |$u$|$u, w$|$u, w, x$|\makecell{$u, w, x, $ \\ $v$}|\makecell{$u, w, x, $ \\ $v, \lnot y$}|\makecell{$u, w, x, $ \\ $v, \lnot y, \lnot z$}}
		\dpllClause{1}{$u, v$}{$u, v$|\done|\done|\done|\done|\done|\done}
		\dpllClause{2}{$\lnot u, w$}{$\lnot u, w$|$w$|\done|\done|\done|\done|\done}
		\dpllClause{3}{$v, \lnot x$}{$v, \lnot x$|$v, \lnot x$|$v, \lnot x$|$v$|\done|\done|\done}
		\dpllClause{4}{$w, y$}{$w, y$|$w, y$|\done|\done|\done|\done|\done}
		\dpllClause{5}{$\lnot v, \lnot y$}{$\lnot v, \lnot y$|$\lnot v, \lnot y$|$\lnot v, \lnot y$|$\lnot v, \lnot y$|$\lnot y$|\done|\done}
		\dpllClause{6}{$\lnot w, z$}{$\lnot w, z$|$\lnot w, z$|$z$|$z$|$z$|$z$|\conflict}
		\dpllClause{7}{$\lnot z, x$}{$\lnot z, x$|$\lnot z, x$|$\lnot z, x$|\done|\done|\done|\done}
		\dpllClause{8}{$y, \lnot z$}{$y, \lnot z$|$y, \lnot z$|$y, \lnot z$|$y, \lnot z$|$y, \lnot z$|$\lnot z$|\done}
		\dpllClause{9}{$x, \lnot u$}{$x, \lnot u$|$x$|$x$|\done|\done|\done|\done}
		\dpllClause{10}{$u$}{$u$|\done|\done|\done|\done|\done|\done}
		\dpllBCP{$u$|$w$|$x$|$v$|$\lnot y$|$\lnot z$| - }
		\dpllPL{ - | - | - | - | - | - | - }
		\dpllDeci{ - | - | - | - | - | - |UNSAT}
	\end{dplltabular}
}

Conflict in step 13\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 0 -> 8
  \draw [->] (1.0602bp,31.0bp) .. controls (2.7468bp,31.0bp) and (39.52bp,31.0bp)  .. (73.786bp,31.0bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (37.5bp,38.5bp) node {$$10$$};
  % Edge: 2 -> 5
  \draw [->] (182.39bp,50.918bp) .. controls (203.43bp,50.753bp) and (251.37bp,50.375bp)  .. (288.82bp,50.08bp);
  \draw (214.0bp,57.5bp) node {$$6$$};
  % Edge: 3 -> 4
  \draw [->] (268.3bp,11.0bp) .. controls (281.47bp,11.0bp) and (304.58bp,11.0bp)  .. (331.88bp,11.0bp);
  \draw (300.0bp,18.5bp) node {$$5$$};
  % Edge: 4 -> 6
  \draw [->] (354.3bp,11.0bp) .. controls (367.47bp,11.0bp) and (390.58bp,11.0bp)  .. (417.88bp,11.0bp);
  \draw (386.0bp,18.5bp) node {$$8$$};
  % Edge: 5 -> 1
  \draw [->] (311.0bp,49.131bp) .. controls (335.99bp,46.927bp) and (403.17bp,40.812bp)  .. (459.0bp,34.0bp) .. controls (461.51bp,33.694bp) and (464.14bp,33.349bp)  .. (476.9bp,31.544bp);
  % Edge: 6 -> 1
  \draw [->] (439.67bp,14.223bp) .. controls (447.24bp,16.746bp) and (458.02bp,20.339bp)  .. (477.09bp,26.697bp);
  % Edge: 7 -> 3
  \draw [->] (182.3bp,11.0bp) .. controls (195.47bp,11.0bp) and (218.58bp,11.0bp)  .. (245.88bp,11.0bp);
  \draw (214.0bp,18.5bp) node {$$3$$};
  % Edge: 8 -> 2
  \draw [->] (95.966bp,33.373bp) .. controls (109.23bp,36.53bp) and (132.98bp,42.186bp)  .. (160.05bp,48.63bp);
  \draw (128.0bp,50.5bp) node {$$2$$};
  % Edge: 8 -> 7
  \draw [->] (95.111bp,26.371bp) .. controls (100.48bp,23.828bp) and (107.48bp,20.828bp)  .. (114.0bp,19.0bp) .. controls (125.65bp,15.733bp) and (139.06bp,13.758bp)  .. (159.92bp,11.639bp);
  \draw (128.0bp,26.5bp) node {$$9$$};
  % Node: 8
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (85.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (85.0bp,31.0bp) node {$u$};
\end{scope}
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (488.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (488.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (171.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (171.0bp,51.0bp) node {$w$};
\end{scope}
  % Node: 5
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (300.0bp,50.0bp) ellipse (11.0bp and 11.0bp);
  \draw (300.0bp,50.0bp) node {$z$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (257.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (257.0bp,11.0bp) node {$v$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (343.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (343.0bp,11.0bp) node {$\lnot y$};
\end{scope}
  % Node: 6
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (429.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (429.0bp,11.0bp) node {$\lnot z$};
\end{scope}
  % Node: 7
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (171.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (171.0bp,11.0bp) node {$x$};
\end{scope}
%
\end{tikzpicture}


}

\scalebox{0.95}{
  \parbox{1cm}{
    \begin{prooftree}
      \AxiomC{$8. \; y \lor \lnot z$}
      \AxiomC{$5. \; \lnot v \lor \lnot y$}
      \BinaryInfC{$\lnot z \lor \lnot v$}
      \AxiomC{$3. \; v \lor \lnot x$}
      \BinaryInfC{$\lnot z \lor \lnot x$}
      \AxiomC{$6. \; \lnot w \lor z$}
      \BinaryInfC{$\lnot x \lor \lnot w$}
      \AxiomC{$2. \; \lnot u \lor w$}
      \BinaryInfC{$\lnot x \lor \lnot u$}
      \AxiomC{$9. \; x \lor \lnot u$}
      \BinaryInfC{$\lnot u$}
      \AxiomC{$10. \; u$}
      \BinaryInfC{$\bot$}
    \end{prooftree}
  }
}