A class of circuits of functional elements over the standard basis of the conjunction, disjunction, and negation elements is considered. For each circuit Σ in this class, its depth D(Σ) and dimension R(Σ) equal to the minimum dimension of the Boolean cube allowing isomorphic embedding Σ are defined. It is established that for n = 1, 2,… and an arbitrary Boolean function f of n variables there exists a circuit Σ_f for implementing this function such that R(Σ_f) ⩽ n − log₂ log₂n + O(1) and D(Σ_f) ⩽ 2n − 2 log₂ log₂n + O(1). It is proved that for n = 1, 2,… almost all functions of n variables allow implementation by circuits of the considered type, whose depth and dimension differ from the minimum values of these parameters (for all equivalent circuits) by no more than a constant and asymptotically no more than by a factor of 2, respectively.

中文翻译：

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关于实现布尔函数的布尔立方体电路中嵌入的复杂性和深度

考虑了基于合取，取反和取反元素的标准基础上的功能元素的电路类别。对于此类中的每个电路Σ，定义其深度D（Σ）和尺寸R（Σ）等于允许同构嵌入Σ的布尔立方体的最小尺寸。已经确定，对于Ñ = 1，2，...和一个任意的布尔函数˚F的Ñ变量存在一个电路Σ _˚F用于实现该功能，使得- [R（Σ _˚F）⩽ ñ -日志₂日志₂ Ñ + Ö（1 ）和D（Σ_f）2 n − 2 log ₂ log ₂ n + O（1）。事实证明，对于n = 1、2，…， n个变量的几乎所有功能都可以由所考虑类型的电路实现，其深度和尺寸与这些参数的最小值（对于所有等效电路）相差不超过a。常数和渐近性分别不超过2倍。