Structural Properties of Invariant Dual Subspaces of Boolean Networks

In this paper, we obtain the following results on dual subspaces of Boolean networks (BNs). For a BN, there is a one-to-one correspondence between partitions of its state-transition graph (STG) and its dual subspaces (i.e., the subspaces generated by a number of Boolean functions of the BN's variables). Moreover, a dual subspace is invariant if and only if the corresponding partition is equitable, i.e., for every two (not necessarily different) cells of the partition, every two states in the former have equally many out-neighbors in the latter. With the help of equitable partitions of an STG, we study the structural properties of the smallest invariant dual subspaces containing a number of Boolean functions. And then, we give algorithms for computing the equitable partition corresponding to the smallest invariant dual subspace containing a given dual subspace. Moreover, we reveal that the unobservable subspace of a BN is the smallest invariant dual subspace containing the output function. We analyze properties of the unobservable subspace by using the obtained structural properties. The graphical representation provides an easier and more intuitive way to characterizing the (smallest) invariant dual subspaces of a BN.