Boole函数的线性可分性(Ⅰ)——n-维超立方体的基本理论

LINEAR SEPARABILITY OF BOOLEAN FUNCTIONS (I)——THE BASIC THEORY OF N-DIMENSIONAL HYPERCUBES

Boole函数的线性可分性是前向人工神经网络理论中的一个比较困难的问题之一。目前仅对变量数n≤7的某些问题给予讨论。当n≥8时,尚无判别Boole函数线性可分的一般准则,更无线性可分Boole函数的计数公式。基于此,本文详细地研究了与Boole函数线性可分性有关的n-维超立方体的基本理论,包括n-维超立方体的基本性质、超立方体中的平行线、子超立方体的计数等,并给出了构造n-维超立方体图的一种新方法。

The linear separability of Boolean functions is one of the difficult problems in the theory of feedforward artificial neural networks. There is no general judgement rule for linear separability of Boolean funtions when n ≥ 8, and no enumeration of the functions. In this paper, the basic properties of n-dimensional hypercube such as the parallel lines in hypercube, the enumeration of sub-hypercube and so on are discussed in detail, and a new method of constructing n-dimensional hypercube graphs is persented.