  We propose a Dynamical Boolean Network (DBN), which is a Virtual Boolean
Network (VBN) whose set of states is fixed but whose transition matrix can
change from one discrete time step to another. The transition matrix $T_{k}$ of
our DBN for time step $k$ is of the form $Q^{-1}TQ$, where $T$ is a transition
matrix (of a VBN) defined at time step $k$ in the course of the construction of
our DBN and $Q$ is the matrix representation of some randomly chosen
permutation $P$ of the states of our DBN. For each of several classes of such
permutations, we carried out a number of simulations of a DBN with two nodes;
each of our simulations consisted of 1,000 trials of 10,000 time steps each. In
one of our simulations, only six of the 16 possible single-node transition
rules for a VBN with two nodes were visited a total of 300,000 times (over all
1,000 trials). In that simulation, linearity appears to play a significant role
in that three of those six single-node transition rules are transition rules of
a Linear Virtual Boolean Network (LVBN); the other three are the negations of
the first three. We also discuss the notions of a Probabilistic Boolean Network
and a Hidden Markov Model--in both cases, in the context of using an arbitrary
(though not necessarily one-to-one) function to label the states of a VBN.
