Improved Lower Bounds for Monotone q-Multilinear Boolean Circuits

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 $\Omega(\frac p {q^4})$ 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 $\Sigma_3$ monotone Boolean circuit for $Isol_{k,n}$ has an exponential in $n$ size or it is not $(k-1)$-multilinear.