The Complexity of 3SAT_N and the P versus NP Problem

TL;DR

This paper introduces a new NP-complete problem 3SAT_N, extends existing classification theorems, and uses novel methods to prove that P does not equal NP, advancing understanding of computational complexity.

Contribution

It defines 3SAT_N, generalizes truth assignments, and employs a combination of algorithms and diagonalization to prove P != NP.

Findings

Introduction of 3SAT_N as an NP-complete problem

Generalization of truth assignments to aggressive truth assignments

Proof that P != NP

Abstract

We introduce the NP-complete problem 3SAT_N and extend Tovey's results to a classification theorem for this problem. This theorem leads us to generalize the concept of truth assignments for SAT to aggressive truth assignments for 3SAT_N. We introduce the concept of a set compatible with the P and NP problem, and prove that all aggressive truth assignments are pseudo-algorithms. We combine algorithm, pseudo-algorithm and diagonalization method to study the complexity of 3SAT_N and the P versus NP problem. The main result is P != NP.