Questionnaire-2024/sat_solvers_dpll/0048_sol.tex

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

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