Some Basic Problems Of Singular Multivalued Logical Networks:Semi-tensor Product Method

In recent years, with the development of science and technology, multi-valued logical networks are extensively studied. As a natural generalization of Boolean networks, multivalued logical networks play important roles in com-puter science, artificial intelligence and complex neural networks. In this paper, using a new operation tool called semi-tensor product of matrices, the descrip-tion and normalization of singular Boolean networks are extended to singular multivalued logical networks.First, we established two kinds of the condensed algebraic expression to describe the dynamic multivalued logical networks with static constraints as for the singular multivalued logical networks. Then, based on the two algebraic forms, new necessary and sufficient conditions are obtained to address the normalization problem of singular k-valued logical networks. Furthermore, existence and uniqueness of the solutions are discussed for singular k-valued logical networks, and the concept of admissible initial values under distinct normalization conditions is presented. Finally, we also discussed concept and solution of the fixed points and cycles of singular k-valued logical networks.