Towards P = NP via k-SAT: A k-SAT Algorithm Using Linear Algebra on Finite Fields

TL;DR

This paper proposes a novel linear algebra-based algorithm for solving k-SAT problems, claiming it may imply P=NP by demonstrating polynomial-time solvability with high probability.

Contribution

It introduces a new k-SAT solving method using linear algebra over finite fields, suggesting potential polynomial-time solutions for NP-complete problems.

Findings

Algorithm runs in roughly O(n^3) time and space.

High probability of correctly identifying satisfiable formulas.

Evidence suggests P=NP based on the proposed method.

Abstract

The problem of P vs. NP is very serious, and solutions to the problem can help save lives. This article is an attempt at solving the problem using a computer algorithm. It is presented in a fashion that will hopefully allow for easy understanding for many people and scientists from many diverse fields. In technical terms, a novel method for solving k-SAT is explained. This method is primarily based on linear algebra and finite fields. Evidence is given that this method may require rougly O(n^3) time and space for deterministic models. More specifically the algorithm runs in time O(P V(n+V)^2) with mistaking satisfiable Boolean expressions as unsatisfiable with an approximate probablity 1 / \Theta(V(n+V)^2)^P, where n is the number of clauses and V is the number of variables. It's concluded that significant evidence exists that P=NP. There is a forum devoted to this paper at http://482527.ForumRomanum.com. All are invited to correspond here and help with the analysis of the algorithm. Source code for the associated algorithm can be found at https://sourceforge.net/p/la3sat.