Singular Boolean networks: Semi-tensor product approach

Singular Boolean networks are introduced in this paper. Via semi-tensor product of matrices and the matrix expression of logical functions, two kinds of the condensed algebraic expressions of singular Boolean networks are obtained. The normalization problem of singular Boolean networks is addressed; that is, under what condition singular Boolean networks can be converted into normal Boolean networks with algebraic restrictions. Then one sufficient condition and one necessary and sufficient condition are derived for the normalization problem. Furthermore, the solvability of singular Boolean networks is discussed and the concept of admissible initial values of singular Boolean networks is presented. Finally, fixed points and cycles of singular Boolean networks are also investigated.