To understand the system function-structure under different background relations, two analysis methods are proposed based on space fault tree and factor space theory. They are the system function-structure minimal disjunctive formula and the system function-structure simplest formula. The former establishes the axiom system of system function-structure analysis based on factor logic, and proves that the system function-structure obtained by function-structure analysis method is a minimal disjunction formula. The latter gives a more detailed analysis steps based on the former idea, which is mainly used to determine the implicit relation between system function and component function. The background sets consisting of the 16 and 32 background relations are studied with these two methods. The results show that the complete background relations can obtain a certain system function-structure uniquely; the incomplete background relations can obtain a family of certain system function-structures. If the incomplete background relations are a subset of the complete background relations, then the incomplete background relations must has a linear relation between functions to complement the incomplete background relations. At the same time, the system function equivalent and replacement relation are obtained. It is also proved that the system function-structure obtained from several subsets of the background sets is the simplest and the condition required for the simplest structure of the background set is strict. At present, the above methods are mainly used for system reliability analysis in safety system engineering.

为了理解不同背景关系下的系统功能结构，基于空间故障树和因子空间理论，提出了两种分析方法。它们是系统功能结构最小析​​取公式和系统功能结构最简单公式。前者建立了基于因子逻辑的系统功能结构分析公理系统，并证明通过功能结构分析方法得到的系统功能结构是最小析取公式。后者基于前者的思想给出了更详细的分析步骤，主要用于确定系统功能和组件功能之间的隐式关系。使用这两种方法研究了由16个背景关系和32个背景关系组成的背景集。结果表明，完整的背景关系可以唯一地获得一定的系统功能结构。不完整的背景关系可以获得一系列特定的系统功能结构。如果不完全背景关系是完全背景关系的子集，则不完全背景关系必须在函数之间具有线性关系以补充不完全背景关系。同时，获得了系统功能的等效关系和置换关系。还证明了从背景集的几个子集获得的系统功能结构是最简单的，并且对于背景集的最简单结构所要求的条件是严格的。目前，以上方法主要用于安全系统工程中的系统可靠性分析。