Detectability vverification of probabilistic Boolean networks

We in this paper study the current-state estimation problem of probabilistic Boolean networks with output observations. First, we propose the concepts of three fundamental categories of detectability in the context of PBNs based on the different purposes, which are periodic detectability, (periodic) k-detectability, and (periodic) d-detectability. Second, utilizing the semi-tensor product technique, we create a uniform methodology for the verification of all the aforementioned categories of detectability. Accordingly, several necessary and sufficient verification criteria are derived. All results obtained in this paper are numerically tractable since they avoid the graph-based symbolic manipulations. Finally, several examples modeled by the same PBN are provided to instantiate the correctness and effeteness of the obtained results.