Dynamics of synchronous Boolean networks with non-binary states.

This article focuses on extending the study of synchronous Boolean networks (BNs) from binary-state variables to networks whose state variables take values in a general Boolean algebra with \(2^p\) elements, where \(p > 1\). Using the Stone representation theorem, the authors establish a framework that represents each non-binary state as a \(p\)-tuple of binary values, enabling the analysis of the dynamics—such as fixed points, periodic orbits, predecessors, garden-of-eden (GOE) configurations, and transients—by relating them to corresponding binary-state BNs called fibers. Key findings include that synchronous BNs with non-binary states exhibit only fixed points and two-periodic orbits, but unlike the binary case, uniqueness of two-periodic orbits is impossible; moreover, the number of fixed points and two-periodic orbits grows combinatorially with \(p\). The paper also characterizes the existence and number of predecessors and GOE configurations, and shows that convergence properties and transient behaviors in the non-binary setting mirror those in the binary case. These results broaden the theoretical foundation of Boolean networks, facilitating their application to real-world phenomena where entities have multiple discrete states beyond simple binary on–off values.