A Linear Representation of Dynamics of Boolean Networks

TLDR: Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system, and a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is express as a multiple linear mapping.

Abstract: A new matrix product, called semi-tensor product of matrices, is reviewed Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor The corresponding algorithms are developed and used to some examples