The complexity of multilayer $d$-dimensional circuits

TL;DR

This paper investigates the complexity of multilayer $d$-dimensional circuits using $ ext{lambda}$-separable graphs, establishing lower bounds on their Shannon function and demonstrating tight bounds for multidimensional rectangular circuits.

Contribution

It introduces a new lower bound for the Shannon function of multilayer circuits based on $ ext{lambda}$-separable graphs and confirms tight bounds for multidimensional rectangular circuits.

Findings

Established Shannon function lower bounds for multilayer circuits.

Derived bounds specifically for $d$-dimensional $ ext{lambda}$-separable graphs.

Proved asymptotic tightness of bounds for multidimensional rectangular circuits.

Abstract

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 $\max \bigl(\frac{2^n}{n}, \frac{2^n (1 - \lambda)}{\log k} \bigr)$ 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.