Learning Probabilistic Logical Control Networks: From Data to Controllability and Observability

This article studies controllability and observability problems for a class of mixed-valued probabilistic logical control networks (PLCNs). First, PLCN is transformed into the algebraic state-space representation (ASSR)-form by resorting to the semitensor product method. Then, the formulas are presented to calculate the lower and upper bounds of the transition probability matrix, which further derive the controllability and observability criteria. Furthermore, the ASSR-form of a PLCN can be regarded as a Markov decision process. Using the latter framework, we prove the equivalence between the controllability probability and the optimal state-value function, which is an iteration equation. Besides, the parallel extension technique transforms the observability of PLCNs into the set stabilization of an augmented system. The correspondence between observability probability and optimal state-value function is also derived. Afterward, based on the state-value function, the algorithms via the Q-learning technique are exploited to estimate the controllability and observability probabilities along with obtaining the corresponding optimal control sequences. Finally, all the theoretical results are elaborated via a genetic regulatory p53-Mdm2 network.