Polynomial Time Algorithm for Boolean Satisfiability Problem (2310.19833v1)

Abstract: This is the latest in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. In the previous papers, we have proved that the sat CNF problem is polynomially reduced to the problem of finding a special covering for a set under the special decomposition of this set and vice versa. That is, these problems are polinomially equivalent. This means that the problem of finding a special covering for a set under a special decomposition of this set, is an NP-complete problem. We also described algorithmic procedures that determine whether there is a special covering for a set under a special decomposition of this set. In this article we prove that all these algorithmic procedures have polynomial time complexity with respect to the length of input data. In addition, we will describe an algorithm that, given any Boolean function in conjunctive normal form (CNF), determines in polynomial time whether this function is satisfiable. We will prove that the time complexity of this algorithm is bounded by the cube of the length of the input data. Also, if the function is not satisfiable, the algorithm deduces this result noting the reason for this result. We have implemented an algorithm in Python, and successfully tested it on Boolean functions represented in CNF with tens of thousands of variables and tens of thousands of clauses.