P versus NP

EasyChair Preprint no. 3061, version 15

14 pagesDate: October 4, 2020

Abstract

$P$ 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 $NP$? The 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. Another major complexity class is $\textit{P-Sel}$. $\textit{P-Sel}$ is the class of decision problems for which there is a polynomial time algorithm (called a selector) with the following property: Whenever it's given two instances, a $yes"$ and a $no"$ instance, the algorithm can always decide which is the $yes"$ instance. It is known that if $NP$ is contained in $\textit{P-Sel}$, then $P = NP$.

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.

Given two Boolean formulas $\phi_{1}$ and $\phi_{2}$ in $3CNF$ with $n$ variables and $m$ clauses, the combinatorial optimization problem $\textit{SELECTOR-3SAT}$ consists in selecting the formula which is satisfiable, where every clause from $\phi_{1}$ and $\phi_{2}$ can be unsatisfied for some truth assignment. We assume that the formula that is satisfiable has the minimum sum of the weighted densities of states. In this way, we solve $\textit{SELECTOR-3SAT}$ with an exact polynomial time algorithm. We claim this problem is a selector of $3SAT$ and thus, $P = NP$.

Keyphrases: completeness, complexity classes, logarithmic space, one-way, polynomial time, reduction