Optimal-Flip-Based Segmented Reinforcement Learning for Detectability Synthesis of Probabilistic Boolean Networks

State estimation is an important property in the operation of logical dynamic systems, and how to analyze and synthesize this property when the estimation conditions are not satisfied is the focus of the current research. This work addresses the synthesis problem of detectability with probability 1 for probabilistic Boolean networks (PBNs) via flip control and optimal-flip-based segmented reinforcement learning (OFSRL). First, in the framework of the semitensor product (STP), the PBN is transformed into an algebraic form to serve as the structure matrix for OFSRL. Depending on the attractor condition in the state pair, the analysis of detectability synthesis is divided into four different cases, and in each case, flip control is applied to the structure matrix and output matrix as actions in the OFSRL framework. Second, the detectability synthesis problem of the PBN is transformed into a set stabilization problem, and to reduce computational complexity, necessary and sufficient conditions based on the reachable set are introduced as criteria for implementing OFSRL. Furthermore, based on the above structure matrix, actions, and criteria, OFSRL is proposed to address the detectability synthesis problem of PBNs and obtain the optimal flipping sequence. Finally, two numerical simulations are conducted to verify the reliability of the proposed conclusions, and comparisons are made between OFSRL, traditional theoretical methods, and conventional reinforcement learning (RL) algorithms to highlight the advantages of OFSRL.