arXiv subjects

Stepan Margaryan

Publications and source records attributed to Stepan Margaryan.

2 recordsLinked to original sources

arXiv2024

Special Coverings of Sets and Boolean Functions

We will study some important properties of Boolean functions based on newly introduced concepts called Special Decomposition of a Set and Special Covering of a Set. These concepts enable us to study important problems concerning Boolean functions represented in conjunctive normal form including the satisfiability problem. Studying the relationship between the Boolean satisfiability problem and the problem of existence of a special covering for set we show that these problems are polynomially equivalent. This means that the problem of existence of a special covering for a set is an NP complete problem. We prove an important theorem regarding the relationship between these problems. The Boolean function in conjunctive normal form is satisfiable if and only if there is a special covering for the set of clauses of this function. The purpose of the article is also to study some important properties of satisfiable Boolean functions using the concepts of special decomposition and special covering of a set. We introduce the concept of generation of satisfiable function by another satisfiable function by means of admissible changes in the clauses of the function. We will prove that if the generation of a function by another function is defined as a binary relation then the set of satisfiable functions of n variables represented in conjunctive normal form with m clauses is partitioned to equivalence classes In addition, extending the rules of admissible changes we prove that arbitrary two satisfiable Boolean functions of n variables represented in conjunctive normal form with m clauses can be generated from each other.

cs.CC↗

arXiv2023

On a Certain NP-Complete Problem

We intend to create new concepts aimed at finding necessary and sufficient conditions for Boolean satisfiability so that these conditions can be verified in polynomial time. Based on these conditions it will be possible to create an algorithm that determines in polynomial time whether a given Boolean formula represented in conjunctive normal form is satisfiable. The work will consist of three articles. This is the first of a planned series of these articles. In this article we introduce the concept of special decomposition of a set and the concept of special covering for a set under such a decomposition. We formulate the decision problem of existance of a special covering for a set under a special decompostition of this set. In order to determine the complexity class in which this problem is located, we study the relationship between the sat CNF problem and the problem of existence of a special covering for a set under a special decomposition of this set. The article proves that the decidability of the sat CNF problem is polynomially reduced to the problem of the existence of a special covering for a set. It is also proved that the problem of existence of a special covering for a set is polynomially reduced to the decidability of the sat CNF problem. Therefore, the mentioned problems are polynomially equivalent. And then, the problem of existence of a special covering for a set is an NP-complete problem. The conditions for the existence of special coverings for a set under special decompositions of this set will be study in the next articles.

cs.CC↗