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A Mathematical Analysis Of Computational Complexity

3290 words - 14 pages

\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 $ij$,
$2\leq i\leq\mathbf{n}$ and $2\leq j\leq\mathbf{m}$, 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
$\vee,\wedge,\neg,\Rightarrow$ endowed with a enumerable set of
atoms $\mathcal{A}$. The set of literals, $\mathcal{L}$, is the set
$\mathcal{A}\cup\{\neg p|p\in\mathcal{A}\}$. A pair of a literal
together with its negation is called a conjugated pair.

\begin{definition}[SAT]\label{SAT:dfn}
A formula $\chi$ in Boolean Logic is said {\em satisfiable}
if there is a valuation $v$ from the set $\{\top,\bot\}$ onto the set
of atoms of $\chi$ so that $v(\chi)$ is {\bf true}. If no such
valuation exists, that is $v(\chi)$ is {\bf false} for any valuation,
then we say that $\chi$ is {\em unsatisfiable}.
\end{definition}

\begin{definition}\label{CNN&MIN:dfn}
A \twsat\ formula $\Psi$ is a conjunctions of a number, say $\mathbf{s}$, of
a disjunction of at most two literals. We write
\[
\Psi\equiv (l_{1}^{1}\vee l_{1}^{2})\wedge\dots\wedge
(l_{\mathbf{s}}^{1}\vee l_{\mathbf{s}}^{2})\equiv
C_{1}\wedge\dots\wedge C_{\mathbf{s}}
\]

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}

Next, we define {\em cards}, a puzzle that,
as we show here, can be solved polynomially is space and time.

\begin{definition}\label{CARD:dfn}
A card\index{card}, $\mathcal{C}$, consists of an array with
$\mathbf{n}$ rows and $\mathbf{m}$ lines.
Each entry in the array, ${ij}$, $2\leq i\leq n$ and $2\leq j\leq m$
contains a number $k_{ij}\geq 1$ of...

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