Questionnaire-2024/sat_solvers_dpll/ex3_sol.tex

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

Conflict in step 3\\
\scalebox{0.75}{

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 1 -> 3
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  % Edge: 1 -> 4
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  \draw (54.0bp,26.5bp) node {$$5$$};
  % Edge: 3 -> 2
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  % 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 {$b$};
\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 {$\lnot b$};
\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{$1. \; a \lor b$}
	\AxiomC{$5. \; a \lor \lnot b$}
	\BinaryInfC{$a$}
\end{prooftree}

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

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

\begin{tikzpicture}[>=latex,line join=bevel,]
  \pgfsetlinewidth{1bp}
%%
\pgfsetcolor{black}
  % Edge: 0 -> 5
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  \pgfsetstrokecolor{strokecol}
  \draw (33.0bp,38.5bp) node {$$7$$};
  % Edge: 2 -> 1
  \draw [->] (258.67bp,47.438bp) .. controls (266.41bp,44.585bp) and (277.51bp,40.495bp)  .. (296.47bp,33.51bp);
  % Edge: 3 -> 2
  \draw [->] (172.97bp,33.373bp) .. controls (186.23bp,36.53bp) and (209.98bp,42.186bp)  .. (237.05bp,48.63bp);
  \draw (205.0bp,50.5bp) node {$$2$$};
  % Edge: 3 -> 4
  \draw [->] (172.11bp,26.371bp) .. controls (177.48bp,23.828bp) and (184.48bp,20.828bp)  .. (191.0bp,19.0bp) .. controls (202.65bp,15.733bp) and (216.06bp,13.758bp)  .. (236.92bp,11.639bp);
  \draw (205.0bp,26.5bp) node {$$4$$};
  % Edge: 4 -> 1
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  % Edge: 5 -> 3
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  \draw (119.0bp,38.5bp) node {$$3$$};
  % Node: 5
\begin{scope}
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  \pgfsetstrokecolor{strokecol}
  \draw (76.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (76.0bp,31.0bp) node {$a$};
\end{scope}
  % Node: 1
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (307.0bp,30.0bp) ellipse (11.0bp and 11.0bp);
  \draw (307.0bp,30.0bp) node {$\bot$};
\end{scope}
  % Node: 2
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (248.0bp,51.0bp) ellipse (11.0bp and 11.0bp);
  \draw (248.0bp,51.0bp) node {$\lnot b$};
\end{scope}
  % Node: 3
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (162.0bp,31.0bp) ellipse (11.0bp and 11.0bp);
  \draw (162.0bp,31.0bp) node {$\lnot c$};
\end{scope}
  % Node: 4
\begin{scope}
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (248.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  \draw (248.0bp,11.0bp) node {$b$};
\end{scope}
%
\end{tikzpicture}


}
\begin{prooftree}
	\AxiomC{$2. \; \lnot b \lor c$}
	\AxiomC{$4. \; b \lor c$}
	\BinaryInfC{$c$}
	\AxiomC{$3. \; \lnot a \lor \lnot c$}
	\BinaryInfC{$\lnot a$}
	\AxiomC{$7. \; a$}
	\BinaryInfC{$\bot$}
\end{prooftree}