section{Introduction}label{INTR:sec}In this article we propose and solve polynomially in time and space a problem which consists in deciding whether, given an array called {em card} of columns $mathbf{n}$ and rows $mathbf{m}$ and whose entries are denoted by $ij$,$2leq ileqmathbf{n}$ and $2leq jleqmathbf{m}$, contain a finite disjunction set of two literals (chosen from a finite set of atoms), then all possible combinations, $mathbf{n}^{mathbf{m}}$, of conjunctions on unsatisfiable or absent . Our interest is theoretical although parallel computing simulations can be performed using the papers presented here. By demonstrating that decision making for a card can be performed polynomially, we have built a new path to understanding computational complexity questions, and in future work we will establish more bounds on complexity questions. A precise definition of {em algorithm} was given by Alan Turing in 1937 (see citation {AT1937}). A question arises spontaneously: {em What is the computational difficulty to execute some algorithms?} See, in chronological order, cite{RMO1959}, cite{RMO1960}, cite{HS1965} and others. The complexity classification can be found in cite{M87 }. Our goal is to pave a way to fully understand the complexity issues on the polynomial resolution of a seemingly large-dimensional problem.%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%%%%%%%%%%%%%%%%%%%%%%%%% section{Basic definitions}label{BCKGRND:sec}We work with a Boolean language whose basic symbols are $vee,wedge,for example,Right Arrow$ equipped with an enumerable set of $mathcal{A}$ atoms. The set of literals, $mathcal{L}$, is set$mathcal{A}cup{eg p|pinmathcal{A}}$. A pair of li...... middle of paper ......algorithm to decide whether a given closed digraph has at least one antichain compatible digraph.%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%Allowing the set of edges, $E$ can be written as an ordered set,${e_{1},dots, e_{k}}$. As a first step, we load the edge set $Search$ with $E$ and for all $1leq ileq k$, we work with $Search={e_{i},dots,e_{k}}$, if $e_{ i}$ is compatible and complete, the search ends with an output{ f There is a compatible chain}, otherwise loads the possible compatible,$PC$ with $e_{i}$ and egin{enumerate}item For all $e_{ i}in E$, for all $ij>i$, if $e_{i} $ and $e_{j}$ are compatible,; itemend{enumerate}