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.

翻译：我们旨在创造新概念，以寻找布尔可满足性的充要条件，使得这些条件能在多项式时间内得到验证。基于这些条件，将可能构建一个算法，在多项式时间内判定以合取范式表示的给定布尔公式是否可满足。本工作由三篇文章组成，这是该系列的第一篇。本文引入了集合的特殊分解概念以及在此分解下集合的特殊覆盖概念。我们提出了在集合的特殊分解下，判定该集合是否存在特殊覆盖的决策问题。为确定该问题所属的复杂度类，我们研究了sat CNF问题与集合在其特殊分解下是否存在特殊覆盖问题之间的关系。本文证明了sat CNF问题的可判定性可多项式归约为集合是否存在特殊覆盖问题，同时也证明了集合是否存在特殊覆盖问题可多项式归约为sat CNF问题的可判定性。因此，上述问题是多项式等价的。进而，集合是否存在特殊覆盖问题是一个NP完全问题。关于集合在其特殊分解下是否存在特殊覆盖的条件将在后续文章中研究。