Towards an Almost Quadratic Lower Bound on the Monotone Circuit Complexity of the Boolean Convolution

We study the monotone circuit complexity of the so called semi-disjoint bilinear forms over the Boolean semi-ring, in particular the n-dimensional Boolean vector convolution. Besides the size of a monotone Boolean circuit, we consider also the and-depth of the circuit, i.e., the maximum number of and-gates on a path to an output gate, and the monom number of the circuit which is the number of distinct subsets of input variables induced by monoms at the output gates. We show that any monotone Boolean circuit of \(\epsilon \log n\)-bounded and-depth computing a Boolean semi-disjoint form with 2n input variables and q prime implicants has \(\varOmega (q/n^{2\epsilon })\) size. As a corollary, we obtain the \(\varOmega (n^{2-2\epsilon })\) lower bound on the size of any monotone Boolean circuit of so bounded and-depth computing the n-dimensional Boolean vector convolution. Furthermore, we show that any monotone Boolean circuit of \(2^{n^{\epsilon }}\)-bounded monom number, computing a Boolean semi-disjoint form on 2n variables, where each variable occurs in p prime implicants, has \(\varOmega (n^{1-2\epsilon }p)\) size. As a corollary, we obtain the \(\varOmega (n^{2-2\epsilon })\) lower bound on the size of any monotone Boolean circuit of \(2^{n^{\epsilon }}\)-bounded monom number computing the n-dimensional Boolean vector convolution. Finally, we demonstrate that in any monotone Boolean circuit for a semi-disjoint bilinear form with q prime implicants that has size substantially smaller than q, the majority of the terms at the output gates representing prime implicants have to have very large length (i.e., the number of variable occurrences). In particular, in any monotone circuit for the n-dimensional Boolean vector convolution of size \(o(n^{2-4\epsilon }/\log n)\) almost all prime implicants of the convolution have to be represented by terms at the circuit output gates of length at least \(n^{\epsilon }\).