Efficient Verification of Observability and Reconstructibility for Large Boolean Control Networks With Special Structures

Verifying observability and reconstructibility of Boolean control networks (BCNs) is NP-hard in the number of nodes. A BCN is observable (reconstructible) if one can use an input sequence and the corresponding output sequence to determine the initial (current) state. In this article, we study when a node aggregation approach can be used to overcome the computational complexity in verifying these properties. We first define a class of node aggregations with subnetworks being BCNs. For acyclic node aggregations in this class, all corresponding subnetworks being observable (reconstructible) implies that the whole BCN is observable (reconstructible), although the converse is not true. In general, for cyclic node aggregations, the whole BCN being observable (reconstructible) does not imply that all subnetworks are observable (reconstructible), or vice versa. We design an algorithm to search for all acyclic node aggregations in this class, and show that finding acyclic node aggregations with small subnetworks can significantly reduce the computational complexity in verifying observability (reconstructibility). We also define a second class of node aggregations with subnetworks being finite-transition systems (more general than BCNs), which compensates for the drawback of the first class when the BCN has only a small number of output nodes. Finally, we use a BCN T-cell receptor kinetics model from the literature with 37 state nodes and 3 input nodes to illustrate the efficiency of the results derived from the two node aggregation methods. For this model, we derive the unique minimal set of 16 state nodes needed to be directly measured to make the overall BCN observable. We also compute 5 of the 16 state nodes needed to be directly measured to make the model reconstructible.