Decomposition of Boolean networks: An approach to modularity of biological systems

The paper proves (i) every Boolean network with a well-defined restriction (e.g., fanout-free) decomposes uniquely (up to Q-consistent reordering) into a semi-direct product of simple networks obtained from the SCCs of its wiring diagram (Theorem 4.3), and (ii) for a two-block decomposition F = F1 oP F2, the attractors satisfy D(F) = ⊔_{C1∈D(F1)} C1 ⊕ D(F2^{C1}), where F2^{C1} is a non-autonomous system driven by a periodic representative of C1 (Theorem 4.8). These are precisely the statements and constructions used in the candidate solution: it builds the product along a topological ordering of SCCs using restrictions to define the couplings, and derives the attractor decomposition via the skew-product viewpoint and periodic forcing. The solution’s arguments align closely with the paper’s definitions and proofs for restriction, semi-direct product, non-autonomous dynamics, and the disjoint union claim for D(F) (see Definition 3.13; Theorem 4.1/4.3; Definition 4.5; Theorem 4.8). Minor differences are present only in exposition (e.g., the model invokes the functoriality of restriction, which is consistent with the paper’s restriction construction but not stated verbatim). Overall, both are correct and essentially the same proof structure.