A logical function can be used to characterize a property of states of a Boolean network (BN), which is considered as an aggregation of states. The dynamics of a set of logical functions are called the dual dynamics of the set. To illustrate the dual dynamics of a given set, which characterizes our concerned properties of a BN, the invariant subspace containing the set of logical functions is proposed, and its properties are investigated. Then, the invariant subspace of Boolean control network (BCN) is also proposed, and its dynamics are obtained. Finally, using outputs as the set of logical functions, the minimum output based dual dynamics is considered and proposed as the minimum realization of BCNs. The minimum realization might have much smaller size, which provides a possible solution to overcome the computational complexity of large scale BNs/BCNs. As an example, the proposed approaches for both BN and BCN are applied to an opinion dynamic network to demonstrate the efficiency of the technique proposed in this article. 

　　Publication: 

　　IEEE Transactions on Automatic Control (Volume: 68, Issue: 4, April 2023) 

　　Author: 

　　Daizhan Cheng 

　　Center of STP Theory and Applications, Liaocheng University, Liaocheng, China 

　　Institute of Systems Science, Chinese Academy of Sciences, Beijing, China 

　　Email: dcheng@iss.ac.cn 

　　Lijun Zhang 

　　School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, China 

　　Dongyao Bi 

　　School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, China