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

摘要

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 log n conjunction-depth computing the n -dimensional Boolean vector convolution has ( n 2 ? 4 ) 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 log n negation-dependent conjunction-depth computes the n n Boolean matrix product then the circuit has ( n 3 ? 2 ) 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.