Exact enumeration of fixed points of AND-OR-NAND-NOR Boolean networks

Juan A. Aledo, Jose P. Llano and Jose C. Valverde

Chaos, Solitons & Fractals, 2025, vol. 195, issue C

Abstract: In this paper, we determine the number of fixed points in Boolean networks where each node has an independent local updating Boolean function in the set {AND,OR,NAND,NOR}. First, we introduce the Sequencing Method to transform a (complex) network into a simpler one with the same fixed points. We use it to convert (non-homogeneous) AND-OR Boolean networks into homogeneous OR (bipartite) Boolean networks with the same number of fixed points, which are then counted exactly. Furthermore, we characterize the existence of fixed points in AND-OR-NAND-NOR Boolean networks. Moreover, based on the results for AND-OR Boolean networks, we provide a formula to count the number of fixed points in AND-OR-NAND-NOR Boolean networks. These results apply to both synchronous and asynchronous (deterministic) update schedules. They are first obtained for binary Boolean networks and then extended to the case of non-binary (or multi-valued) Boolean networks. The relevance of the networks studied makes our work highly valuable to model computational processes or other real-world phenomena.

Keywords: Boolean networks; Fixed points; Exact enumeration problems (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:195:y:2025:i:c:s096007792500195x

DOI: 10.1016/j.chaos.2025.116182

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