A directed projection graph of the n-dimensional hypercube on the two-dimensional plane is successfully created. Any n-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on n-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of n-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.

中文翻译：

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平衡线性可分布尔函数的N维超立方和亚超立方分解的有向投影图

的有向投影图n二维平面上的一维超立方体创建成功。任何n-变量布尔函数可以很容易地转换为投影的诱导子图。因此，关于n- 变量布尔函数只需要关注一个二维平面图。建立了投影图和诱导子图的一些数学理论，揭示了平衡线性可分布尔函数（BLSBF）的一些性质和特征。特别是 BLSBF 的子超立方分解很容易在投影上表示，同时，计数的枚举方案n-变量 BLSBFs 是通过使用等价分类和保形变换开发的。借助本文构建的有向投影图，可以进一步研究布尔函数、人工神经网络等领域的许多难题。