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

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.

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