We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most $ϵ\log n$ conjunction-depth computing the n-dimensional Boolean vector convolution has $\mathrm{\Omega }(n^{2-4ϵ})$ and-gates. For Boolean matrix product, we derive even a stronger lower-bound trade-off. Instead of conjunction-depth we use the negation-dependent conjunction-depth, where one counts only and-gates whose each direct predecessor has a (not necessarily direct) predecessor representing a negated input variable. We show that if a normalized Boolean circuit of at most $ϵ\log n$ negation-dependent conjunction-depth computes the $n\times n$ Boolean matrix product then the circuit has $\mathrm{\Omega }(n^{3-2ϵ})$ and-gates. We complete our lower-bound trade-offs for the Boolean convolution and matrix product with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.

我们考虑归一化布尔电路，该布尔电路使用取和与​​合的二元运算以及一元求反，但有一个限制，即求反只能应用于输入变量。我们在计算布尔半不相交双线性形式的归一化布尔电路的大小与它们的联合深度（即，到输出门的有向路径上的“与”门的最大数量）之间得出一个下限权衡。特别是，我们表明，最多不存在任何归一化布尔电路$ϵ\mathrm{日志}ñ$联合深度计算n维布尔向量卷积具有$\mathrm{\Omega }（{ñ}^{2-4ϵ}）$和门。对于布尔矩阵乘积，我们得出了更强的下界权衡。代替依存深度，我们使用与否定相关的依存深度，其中仅计算和门，其每个直接前任都有一个表示负输入变量的（不一定是直接）前任。我们证明，如果归一化布尔电路最多为$ϵ\mathrm{日志}ñ$ 依赖于否定的合计深度计算 $ñ\times ñ$ 布尔矩阵乘积则电路具有 $\mathrm{\Omega }（{ñ}^{3-2ϵ}）$和门。我们用已知的快速代数算法产生的相似形式的上限折衷来完成布尔卷积和矩阵乘积的下限折衷。