Does NP not equal P?

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century. Second, building on the theory of functional relations in mathematics, it provides a theoretical framework where we can rigorously distinguish two pairs of concepts: Between solving a problem and verifying the solution to a problem. Between a d...

A comprehensive review on Cook's contribution in the theory of NP-Completeness with relations to modern mathematics.

$P \overset{\text{?}}{=} NP$ or $P\ vs\ NP$ is the core problem in computational complexity theory. In this paper, we proposed a definition of linear correlation of derived matrix and system, and discussed the linear correlation of $P$ and $NP$. We draw a conclusion that $P$ is linearly dependent and there exists $NP$ which is is linearly independent and take a 3SAT instance which belongs to $NP$ as the example , that is, $P \neq NP$.

We claim to resolve the P=?NP problem via a formal argument for P=NP.

In this paper, we make a preliminary interpretation of Cook's theorem presented in [1]. This interpretation reveals cognitive biases in the proof of Cook's theorem that arise from the attempt of constructing a formula in CNF to represent a computation of a nondeterministic Turing machine. Such cognitive biases are due to the lack of understanding about the essence of nondeterminism, and lead to the confusion between different levels of nondeterminism and determinism, thus cau...

The $\textbf{P}$ vs. $\textbf{NP}$ problem is an important problem in contemporary mathematics and theoretical computer science. Many proofs have been proposed to this problem. This paper proposes a theoretic proof for $\textbf{P}$ vs. $\textbf{NP}$ problem. The central idea of this proof is a recursive definition for Turing machine (shortly TM) that accepts the encoding strings of valid TMs. By the definition, an infinite sequence of TM is constructed, and it is proven that ...

Two theorems about the P versus NP problem be proved in this article (1) There exists a language $L$, that the statement $L \in \textbf{P}$ is independent of ZFC. (2) There exists a language $L \in \textbf{NP}$, for any polynomial time deterministic Turing machine $M$, we cannot prove $L$ is decidable on $M$.

In this paper, we prove that no deterministic algorithm can solve SAT in polynomial time in the number of boolean variables.

Theoretical complexity is a vital subfield of computer science that enables us to mathematically investigate computation and answer many interesting queries about the nature of computational problems. It provides theoretical tools to assess time and space requirements of computations along with assessing the difficultly of problems - classifying them accordingly. It also garners at its core one of the most important problems in mathematics, namely, the $\textbf{P vs. NP}$ mil...

The relationship between the complexity classes P and NP is an unsolved question in the field of theoretical computer science. In this paper, we look at the link between the P - NP question and the "Deterministic" versus "Non Deterministic" nature of a problem, and more specifically at the temporal nature of the complexity within the NP class of problems. Let us remind that the NP class is called the class of "Non Deterministic Polynomial" languages. Using the meta argument t...