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.

翻译：单调布尔电路由或门、与门以及与输入变量和布尔常量对应的输入门组成。若对于其每个输出门o以及该门所计算函数的每个素蕴含项s，将或门和与门分别替换为加法门和乘法门后得到的算术版本电路，在o处计算的多项式中包含一个单项式，该单项式包含与s相同的变量，且s中的每个变量在该单项式中的次数至多为q，则该电路称为q-多重线性的。首先，我们研究基于单调q-多重线性布尔电路规模计算半不交双线性布尔形式的复杂度。特别地，我们证明任何计算包含p个素蕴含项的半不交布尔形式的单调1-多重线性布尔电路至少包含p个与门。我们还证明任何计算包含p个素蕴含项的半不交布尔形式的单调q-多重线性布尔电路的规模为Ω(p/q⁴)。其次，我们研究基于单调q-多重线性布尔电路验证k维布尔矩阵每行（例如k=2时每行每列）至少有一个1的单调布尔函数Isol_{k,n}的复杂度。我们证明任何用于Isol_{k,n}的Σ₃单调布尔电路要么规模关于n指数增长，要么不是(k-1)-多重线性的。