Solvability and Normalization of General Singular Boolean Networks via Admissible and Normal Initial State Sets

This article addresses the fundamental issues of solvability and normalization in singular Boolean networks (SBNs) from a new perspective based on the admissible and normal initial state sets. It presents novel results and removes the restrictive conditions found in existing literature. First, the state transition matrix of SBNs is constructed by defining a new operator, and the admissible initial state set with the normal initial state set, of SBNs is introduced. Second, the problems of the solvability and uniqueness of the solution to a general SBN are converted into computing its admissible and normal initial state sets, which can be analytically computed using the derived formulas. Third, based on the normal initial state set, a necessary and sufficient condition for solving the normalization problem of general SBNs is established for the first time, which removes the restrictions in the existing literature. Finally, the results obtained are compared with existing literature and illustrated with examples.