A Mathematical Analysis of Computational Complexity

section{Introduction}label{INTR:sec}

In this paper we propose and solve polynomially in time and space a problem that consists on deciding whether, given an array called {em card} of $mathbf{n}$ columns and

$mathbf{m}$ lines and whose entries, denoted $ij$,

$2leq ileqmathbf{n}$ and $2leq jleqmathbf{m}$, contain a finite set of disjunction of two literals (chosen among a finite set of atoms), then all possible combinations, $mathbf{n}^{mathbf{m}}$, of conjunctions over unsatisfiable or no. Our interest is theoretical although simulations of parallel computation can be performed using cards introduce here. On showing that the decision process for a card can be performed polynomially, we built a new path to understand computational complexity questions and in future works, we will establish more bounds in complexity questions.

A precise definition of {em algorithm} was given by Alan Turing in

1937 (see cite{AT1937}). A natural question arises: {em What is the computational difficult to perform some algorithm?} See, in chronological order, cite{RMO1959}, cite{RMO1960}, cite{HS1965} and others.

Classification of complexity are find in cite{M87}. Our aims are to to open a way to deeply understand complexity questions on polynomially solving an apparently expsize problem.

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% section{Basic Definitions}label{BCKGRND:sec}

We work with a Boolean Language whose basic symbols are

$vee,wedge,

eg,Rightarrow$ endowed with a enumerable set of atoms $mathcal{A}$. The set of literals, $mathcal{L}$, is the set

$mathcal{A}cup{

eg p|pinmathcal{A}}$. A pair of a li...

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...gorithm to decide if a given closed digraph has at least one compatible antichain.

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Let the set of edges, $E$ be written as an ordered set,

${e_{1},dots,e_{k}}$.

As a first step, we load the set of edges $Search$ 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}, else, load 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,; item

end{enumerate}