A subformula $C_{k}\equiv l_{k}^{1}\vee l_{k}^{2}$,
$1\leq k\leq \mathbf{s}$ of $\Psi$ is called a {\em clause}.
\end{definition}
We define the puzzle of {\em solving a card} and show that the any card of size $\mathbf{n}=2$ can be solved polynomially in space and time.
\begin{notation}\label{N2M:ntt}
Let $\mathbf{m}^{\mathbf{n}}$, the set of all mappings from $n$ to
$\mathbf{m}$. Write an element of $\mathbf{m}^{\mathbf{n}}$ as
$\{1j_{1},2j_{2},\dots,\mathbf{n}j_{\mathbf{n}}\}$, that is, the above set represents the graphic of a mapping $f:\mathbf{n} \mapsto\mathbf{m}$ and an
{\em entry}\index{entry}, $ij$ has the meaning that $f(i)=j$
\end{notation}
\begin{definition}\label{CARD:dfn}
A card\index{card}, $\mathcal{C}$, consists of an…show more content… The starting point is to define the cylindrical digraph associated with a given card. The vertices of a cylindrical digraph are associated with the disjunction of each entry. A path over the cylindrical digraph is associated to conjugations of a disjunctions in choosen entries.
If $cl\equiv p\vee q$ is a disjunction that belongs, say, to clauses $i_{1}j_{1},\dots,i_{r}j_{r}$, then both
$\neg p\Rightarrow q$ and $\neg q\Rightarrow p$ are vertices of the cylindrical digraph and, moreover, the label associated to both vertices is
$i_{1}j_{1},\dots,i_{r}j_{r}$. The vertices $\neg p\Rightarrow q$ and $\neg q\Rightarrow p$ contains in their set of labels all the entries $ij$ so that $cl\equiv p\vee q$ belongs to $ij$. Some subgraphs of the cylindrical digraph will be associated to unsatisfiable formulas and we keep track of these subgraphs in order to decide, in an optimal way whether they represent or no all possible combinations. \begin{definition}\label{CYL:dfn}
Given a card, $\mathcal{C}$, define the
{\em cylindrical digraph}\index{cylindrical digraph}
$\clndr$-graph$=\langle V,E,Labels\rangle$, a pair digraph and a mapping from $E$ to the set of parts of the…show more content… \end{itemize}
\item If $a\equiv a\vee\bot$ belongs to the set of entries $\{i_{1}j_{1},i_{2}j_{2},\dots,i_{k}j_{k}\}$, then, \begin{itemize} \item $\neg a\Rightarrow \bot$ belongs to the set of vertices, $V$; \item $i_{1}j_{1},i_{2}j_{2},\dots,i_{k}j_{k}$ is the label assigned to $\neg a\Rightarrow\bot$. \end{itemize}
Two vertices of the form $a\Rightarrow b$ and $b\Rightarrow c$ form the edge
\[
(a\Rightarrow b)\mapsto(b\Rightarrow c)
\]
