P versus NP

$P$ versus $NP$ is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is $P$ equal to $NP$? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the $P$ versus $NP$ problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. To attack the $P$ versus $NP$ question the concept of $\textit{NP-completeness}$ has been very useful. A well-known $\textit{NP-complete}$ problem is $3SAT$. In $3SAT$, it is asked whether a given Boolean formula $\phi$ in $3CNF$ is satisfiable.

In this paper, we consider the problem of computing the sum of the weighted densities of states of a Boolean formula in $3CNF$. Given a Boolean formula $\phi$, the density of states $n(E)$ counts the number of truth assignments that leave exactly $E$ clauses unsatisfied in $\phi$. The weighted density of states $m(E)$ is equal to $E \times n(E)$. The sum of the weighted densities of states of a Boolean formula in $3CNF$ with $m$ clauses is equal to $\sum_{E = 0}^{m} m(E)$. We prove that we can calculate the sum of the weighted densities of states in polynomial time.

Diophantine equations of the form $\sum_{E = 0}^{m} E \times x_{E} = c$ are solvable in polynomial time for arbitrary values of $m$. We can apply this Diophantine equation such that $c$ is the value of the sum of the weighted densities of states from a Boolean formula $\phi$ in $3CNF$, $E \times x_{E}$ is the value of each term $E \times n(E)$, where $n(E)$ corresponds to the unknown value of $x_{E}$. Hence, we only need to check that $\sum_{E = 1}^{m} E \times x_{E} = c$ has its positive solutions such that $\sum_{E = 1}^{m} x_{E} \neq 2^{n}$ is always true when $x_{E} > 0$ and $x_{E + 2} > 0$ implies that $x_{E + 1} > 0$ and when $x_{E} > 0$ and $x_{E + 1} = 0$ implies that $x_{E + 2} = 0$, where $n$ is the number of variables in $\phi$.

Certainly, if the Boolean formula $\phi$ has $n$ variables, then $\phi$ has exactly $2^{n}$ possible truth assignments. In this way, we are able to check whether the Boolean formula $\phi$ is satisfiable and obtain a solution for the problem $3SAT$ in polynomial time. If any $\textit{NP-complete}$ problem is in $P$, then every $NP$ problem can be decided by a polynomial time algorithm. Consequently, we show the complexity class $P$ is equal to $NP$.