The complexity of multilayer $d$-dimensional circuits

In this paper we research a model of multilayer circuits with a single logical layer. We consider $ \lambda $-separable graphs as a support for circuits. We establish the Shannon function lower bound $ \max \left(\frac{2^n}{n}, \frac{2^n (1 - \lambda)}{\log k} \right)$ for this type of circuits where k is the number of layers. For $d$-dimensional graphs, which are $ \lambda $-separable for $ \lambda $ = $ \frac{d - 1}{d} $, this gives the Shannon function lower bound $ \frac{2^n}{\min(n, d \log k)} $. For multidimensional rectangular circuits the proved lower bound asymptotically matches to the upper bound.