Necessary and Sufficient Condition for Satisfiability of a Boolean Formula in CNF and its Implications on P versus NP problem

TL;DR

This paper introduces a novel polynomial-time algorithm for Boolean satisfiability in CNF, based on properties of fully populated clauses, suggesting P equals NP, with potential optimizations for improved performance.

Contribution

It presents a new necessary and sufficient condition for CNF satisfiability and a polynomial-time algorithm, challenging the P versus NP problem.

Findings

A relationship between sibling clauses and truth assignments is established.

A polynomial-time satisfiability algorithm is developed.

Optimizations for the algorithm are proposed.

Abstract

Boolean satisfiability problem has applications in various fields. An efficient algorithm to solve satisfiability problem can be used to solve many other problems efficiently. The input of satisfiability problem is a finite set of clauses. In this paper, properties of clauses have been studied. A type of clauses have been defined, called fully populated clauses, which contains each variable exactly once. A relationship between two unequal fully populated clauses has been defined, called sibling clauses. It has been found that, if one fully populated clause is false, for a truth assignment, then all it's sibling clauses will be true for the same truth assignment. Which leads to the necessary and sufficient condition for satisfiability of a boolean formula, in CNF. The necessary and sufficient condition has been used to develop a novel algorithm to solve boolean satisfiability problem in polynomial time, which implies, P equals NP. Further, some optimisations have been provided that can be integrated with the algorithm for better performance.