In this article we introduce the concept of special decomposition of a set and the concept of special covering of a set under such a decomposition. Our goal is to study the conditions for the existence of special coverings for sets under special decompositions of these sets, as well as to study the degree of complexity of finding them. These conditions for the formulated problem will have important applications in the field of satisfiability of Boolean functions. In order to determine the complexity class in which this problem is located, we study the relationship between sat CNF problem and the problem of existence of special covering for the set. We also will study the relationship between classes of computational complexity by searching for special coverings of the sets. 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.

翻译：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。我们的目标是研究在集合特殊分解下存在特殊覆盖的条件，以及寻找这些特殊覆盖的复杂度。该问题所阐述的条件在布尔函数的可满足性领域具有重要应用。为了确定此问题所属的复杂度类，我们研究了SAT-CNF问题与集合特殊覆盖存在性问题之间的关系。同时，我们还通过寻找集合的特殊覆盖来研究计算复杂度类之间的关系。本文证明了SAT-CNF问题的可判定性可多项式归约到集合特殊覆盖的存在性问题，反之，集合特殊覆盖的存在性问题也可多项式归约到SAT-CNF问题的可判定性。因此，上述问题是多项式等价的。进而，集合特殊覆盖的存在性问题是一个NP完全问题。