On Boolean control networks with maximal topological entropy

Boolean control networks (BCNs) are discrete-time dynamical systems with Boolean state-variables and inputs that are interconnected via Boolean functions. BCNs are recently attracting considerable interest as computational models for genetic and cellular networks with exogenous inputs. The topological entropy of a BCN with m inputs is a nonnegative real number in the interval [0,mlog2]. Roughly speaking, a larger topological entropy means that asymptotically the control is "more powerful". We derive a necessary and sufficient condition for a BCN to have the maximal possible topological entropy. Our condition is stated in the framework of Cheng's algebraic state-space representation of BCNs. This means that verifying this condition incurs an exponential time-complexity. We also show that the problem of determining whether a BCN with n state variables and m=n inputs has a maximum topological entropy is NP-hard, suggesting that this problem cannot be solved in general using a polynomial-time algorithm.