Synthesis of Boolean networks via semi-tensor product

This paper gives a systematic survey on the semi-tensor product approach to logical systems such as Boolean (control) networks. First the history of Boolean networks is reviewed, and the mathematical frameworks for Boolean network, Boolean control network, and probabilistic Boolean (control) network respectively are presented. Then the semi-tensor product of matrices is introduced as a fundamental tool of this approach. Using it, two key techniques for this approach: (1) algebraic (or matrix) expression of logic; and (2) the state space description of logical dynamics, are introduced. The topological structure of Boolean network is analyzed via this approach. Finally, some control problems are investigated, which include controllability and observability, realization, disturbance decoupling, stability and stabilization, optimal control, identification, etc. In one word, the paper intends to provide a comprehensive guideline to the theory of synthesis and control of Boolean networks.