Observability of Boolean control networks: A unified approach based on the theories of finite automata and formal languages

In this paper, we solve a basic problem of whether there are algorithms to determine the observability of Boolean control networks (BCNs). In fact, we give a unified approach to design algorithms to judge whether a given BCN is observable with respect to different observability. In this work, an algorithm determining the observability of BCNs is a deterministic mechanical procedure that receives a BCN and after a finite number of processing steps it returns “Yes”, if the BCN is observable; and returns “No”, otherwise. First, we investigate the implication relationship between different observability of BCNs, which are the most general observability, i.e., any two distinct states can be distinguished by a designed control sequence, and the observability proposed in [D. Cheng, H. Qi (2009). Controllability and observability of Boolean control networks, Automatica, 45(7), 1659-1667.] and [D. Cheng, Y. Zhao (2011). Identification of Boolean control networks, Automatica, 47, 702-710.], respectively. Second, we put forward a concept of weighted pair graph for BCNs, using which, based on the theories of finite automata and formal languages, we give equivalent test criteria and further design algorithms to judge whether a given BCN is observable with respect to these observability, respectively, in the framework of the semi-tensor product of matrices.