Cyclic Attractor Detection in Boolean Network Dynamics under Local Logical Constraints

Abstract

Boolean networks are finite discrete nonlinear systems whose long-term behaviour is organised by fixed-point and cyclic attractors. Detecting such recurrent states is important in applications ranging from gene regulation and neural computation to complex-network models, but the computational boundary between tractable and intractable attractor analysis is still not fully understood. We study that boundary from the perspective of local logical rules. We consider Boolean networks under parallel update whose coordinate functions are given by circuits over a fixed finite basis of a closed Boolean-function class, and ask whether the network has a cyclic attractor of prescribed exact period $k$. For every fixed $k\ge 2$, we obtain a complete complexity dichotomy over Post's lattice. The problem is $\mathrm{NP}$-complete whenever the local rule class contains majority-like self-dual rules or one of the two mixed conjunctive-disjunctive monotone families. In all remaining Post classes it is polynomial-time solvable, with affine rules and pure conjunctive or pure disjunctive rules with constants providing the boundary tractable cases. The results show that exact attractor detection is governed not only by the network architecture but also by the logical mechanism of local update: affine and one-sided rules preserve algebraic or order structure, whereas majority-like and mixed monotone rules can encode global Boolean consistency constraints.