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Semi-tensor product of matrices (STP of matrices) is a new matrix product and has been successfully applied to many fields, especially to logical dynamic systems. This paper considers how to represent a Boolean matrix as an exponentiation of another one in the framework of the STP of matrices. To this end, Boolean matrices (BMs) are classified into three types, single block, full block and non-full block BMs. For each kind of BMs, a sufficient and necessary condition for the exponentiation-representability is proposed, and an algebraic algorithm of finding all the exponentiation representations of a Boolean matrix is designed. As potential applications, a discussion of how to use the obtained results to analyze and synthesize logical systems mathematically, especially to finite state machines, is finally presented. The results of this paper may provide a theory basis for studying finite-valued systems, logical systems, logical networks and networked evolutionary games, etc.

Exponentiation Representation of Boolean Matrices in the Framework of Semi-Tensor Product of Matrices