Improved Lower Bounds for Monotone q-Multilinear Boolean Circuits
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A monotone Boolean circuit is composed of OR gates, AND gates and input gates corresponding to the input variables and the Boolean constants. It is q-multilinear if for each its output gate o and for each prime implicant s of the function computed at o, the arithmetic version of the circuit resulting from the replacement of OR and AND gates by addition and multiplication gates, respectively, computes a polynomial at o which contains a monomial including the same variables as s and each of the variables in s has degree at most q in the monomial. First, we study the complexity of computing semi-disjoint bilinear Boolean forms in terms of the size of monotone q-multilinear Boolean circuits. In particular, we show that any monotone 1-multilinear Boolean circuit computing a semi-disjoint Boolean form with p prime implicants includes at least p AND gates. We also show that any monotone q-multilinear Boolean circuit computing a semi-disjoint Boolean form with p prime implicants has Ω(p/q^4) size. Next, we study the complexity of the monotone Boolean function Isol_k,n that verifies if a k-dimensional Boolean matrix has at least one 1 in each line (e.g., each row and column when k=2), in terms of monotone q-multilinear Boolean circuits. We show that that any Σ_3 monotone Boolean circuit for Isol_k,n has an exponential in n size or it is not (k-1)-multilinear.
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