On detectability of probabilistic Boolean networks

Abstract

This paper devotes to introducing and investigating the detectability of probabilistic Boolean networks (PBNs). Two types of detectability, weak detectability and strong detectability, are proposed to deal with different situations. Firstly, by resorting to the semi-tensor product (STP) of matrices, the equivalent algebraic expression of the PBN is converted into another kind of expression, which is called the data form. This data form can estimate the current state distribution from the corresponding output sequence. Then, using this data form constructs the detection form, based on which, two necessary and sufficient conditions are derived for checking weak detectability and strong detectability. Secondly, combining the STP and this detection form, a nice algorithm is established for checking wether a PBN is weak detectability. Finally, for better presentation, two numerical examples are shown to demonstrate the effectiveness of the theoretical results, and the diagnosis of tuberculosis is briefly analyzed.