We consider Boolean networks with interaction graphs partitioned into strongly connected components, which we call strong modules. This type of network decomposition has been considered in the literature, primarily from the perspective of attractor detection algorithms. In this paper, we aim to provide an algebraic basis for this line of research in the case of asynchronous Boolean networks. We prove that the asynchronous attractors of a network can be described as a dependent sum construction: as products of attractors of its controlled strong modules. We then show that a representation of all attractors can be computed in polynomial time under two conditions: the strong modules are small, and either the network is sparse or its defining functions have small size circuits (in particular when they are nested canalizing). We illustrate these results on a published Boolean model.

翻译：本文考虑将交互图划分为强连通分量的布尔网络，并称这些分量为强模块。此类网络分解在已有文献中主要从吸引子检测算法的角度被探讨。本文旨在为非同步布尔网络的相关研究提供代数基础。我们证明，网络的非同步吸引子可描述为一种依赖积构造：即其受控强模块吸引子的乘积。进而证明，在满足两个条件时，所有吸引子的表示可在多项式时间内计算：强模块规模较小，且网络是稀疏的或其定义函数具有小规模电路（特别当函数为嵌套通道化时）。我们通过一个已发表的布尔模型示例验证了这些结果。