Questionnaire-2024/sat_solvers_dpll/0056_sol.tex

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

Conflict in step 4\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 1 -> 3
  \draw [->] (21.966bp,33.373bp) .. controls (35.227bp,36.53bp) and (58.98bp,42.186bp)  .. (86.047bp,48.63bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (54.0bp,50.5bp) node {$$3$$};
  % Edge: 1 -> 4
  \draw [->] (21.111bp,26.371bp) .. controls (26.482bp,23.828bp) and (33.482bp,20.828bp)  .. (40.0bp,19.0bp) .. controls (51.65bp,15.733bp) and (65.057bp,13.758bp)  .. (85.923bp,11.639bp);
  \draw (54.0bp,26.5bp) node {$$4$$};
  % Edge: 3 -> 2
  \draw [->] (107.67bp,47.438bp) .. controls (115.41bp,44.585bp) and (126.51bp,40.495bp)  .. (145.47bp,33.51bp);
  % Edge: 4 -> 2
  \draw [->] (107.67bp,14.223bp) .. controls (115.24bp,16.746bp) and (126.02bp,20.339bp)  .. (145.09bp,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 {$b$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,51.0bp) node {$e$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (97.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (97.0bp,11.0bp) node {$\lnot e$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (156.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (156.0bp,30.0bp) node {$\bot$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$3. \; \lnot b \lor e$}
	\AxiomC{$4. \; \lnot b \lor \lnot e$}
	\BinaryInfC{$\lnot b$}
\end{prooftree}

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

Conflict in step 7\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 0 -> 4
  \draw [->] (1.0536bp,31.0bp) .. controls (2.5071bp,31.0bp) and (33.188bp,31.0bp)  .. (64.696bp,31.0bp);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (33.0bp,38.5bp) node {$$7$$};
  % Edge: 2 -> 1
  \draw [->] (172.67bp,47.438bp) .. controls (180.41bp,44.585bp) and (191.51bp,40.495bp)  .. (210.47bp,33.51bp);
  % Edge: 3 -> 1
  \draw [->] (172.67bp,14.223bp) .. controls (180.24bp,16.746bp) and (191.02bp,20.339bp)  .. (210.09bp,26.697bp);
  % Edge: 4 -> 2
  \draw [->] (86.966bp,33.373bp) .. controls (100.23bp,36.53bp) and (123.98bp,42.186bp)  .. (151.05bp,48.63bp);
  \draw (119.0bp,50.5bp) node {$$5$$};
  % Edge: 4 -> 3
  \draw [->] (86.111bp,26.371bp) .. controls (91.482bp,23.828bp) and (98.482bp,20.828bp)  .. (105.0bp,19.0bp) .. controls (116.65bp,15.733bp) and (130.06bp,13.758bp)  .. (150.92bp,11.639bp);
  \draw (119.0bp,26.5bp) node {$$6$$};
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (76.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (76.0bp,31.0bp) node {$\lnot b$};
\end{scope}
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (221.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (221.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (162.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (162.0bp,51.0bp) node {$f$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (162.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (162.0bp,11.0bp) node {$\lnot f$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$5. \; b \lor f$}
	\AxiomC{$6. \; b \lor \lnot f$}
	\BinaryInfC{$b$}
	\AxiomC{$7. \; \lnot b$}
	\BinaryInfC{$\bot$}
\end{prooftree}