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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 ϵlogn conjunction-depth computing the n-dimensional Boolean vector convolution has ω(n2−4ϵ) and-gates. Analogously, any normalized Boolean circuit of at most ϵlogn conjunction-depth computing the n × n Boolean matrix product has ω(n3−4ϵ) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.

Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth