Necessary and sufficient conditions for Boolean satisfiability

TL;DR

This paper establishes necessary and sufficient algorithmic conditions for Boolean satisfiability by linking it to special set coverings, providing a new perspective on the problem's structure and solutions.

Contribution

It introduces a novel set-theoretic approach to characterize satisfiability, proving the existence of conditions that determine when a Boolean function in CNF is satisfiable.

Findings

Existence of necessary and sufficient conditions for Boolean satisfiability.

Algorithmic procedures to determine satisfiability and variable assignments.

Connection between special set coverings and Boolean satisfiability.

Abstract

This is the second in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. The research in this article aims to find conditions of an algorithmic nature that are necessary and sufficient to transform any Boolean function in conjunctive normal form into a specific form that guarantees the satisfiability of this function. To find such conditions, we use the concept of a special covering of a set introduced in [13], and investigate the connection between this concept and the notion of satisfiability of Boolean functions. As shown, the problem of existence of a special covering for a set is equivalent to the Boolean satisfiability problem. Thus, an important result is the proof of the existence of necessary and sufficient conditions that make it possible to find out if there is a special covering for the set under the special decomposition. This result allows us to formulate the necessary and sufficient algorithmic conditions for Boolean satisfiability, considering the function in conjunctive normal form as a set of clauses. In parallel, as a result of the aforementioned algorithmic procedure, we obtain the values of the variables that ensure the satisfiability of this function. The terminology used related to graph theory, set theory, Boolean functions and complexity theory is consistent with the terminology in [1], [2], [3], [4]. The newly introduced terms are not found in use by other authors and do not contradict to other terms.