Input/output decoupling of Boolean control networks

In this study, the author investigates the input/output decoupling problem for Boolean control networks (BCNs). To keep up with the spirit of the original definition for linear state-space models, that pertains the relationship between inputs and outputs independently of the state variables, two properties that formalise in different ways the idea that each single component of the output depends on the values of the corresponding input are provided, but not on the values of the other inputs. These properties are introduced by referring to the classical representation of a BCN in terms of Boolean input, state and output vectors, whose mutual relationships are expressed through the logical operators AND, OR and so on. In this set-up, it is proven that there is some natural ordering among these properties, namely that one of them implies the other. In the second part of this study, it is shown that by resorting to the algebraic representation of BCNs a complete characterisation of these properties is possible. The algebraic characterisations obtained through this approach provide easy to check algorithms to evaluate whether a BCN is input/output decoupled or not. Finally, graph-theoretic characterisations of the two input/output decoupling properties are provided.