About the impossibility to prove P=NP and the pseudo-randomness in NP

TL;DR

This paper argues that proving P=NP is impossible within a time-independent mathematical framework and explores the time-dependent nature of NP problems linked to pseudo-randomness.

Contribution

It introduces a novel perspective connecting the P vs NP problem to the temporal and pseudo-random aspects of NP problems, arguing the impossibility of proof in a time-independent setting.

Findings

Proves P=NP cannot be established in a time-independent mathematical framework.

Highlights the role of pseudo-randomness in the complexity of NP problems.

Shows the temporal nature of NP problems is essential to understanding their complexity.

Abstract

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 that results in Mathematics should be "time independent" as they are reproducible, the paper shows that the P!=NP assertion is impossible to prove in the a-temporal framework of Mathematics. In a previous version of the report, we use a similar argument based on randomness to show that the P = NP assertion was also impossible to prove, but this part of the paper was shown to be incorrect. So, this version deletes it. In fact, this paper highlights the time dependence of the complexity for any NP problem, linked to some pseudo-randomness in its heart.