A Tutorial Exposition of the Analysis of Synchronous Boolean Networks via Semi-Tensor Products

Abstract

This paper is a detailed tutorial exposition of the analysis of synchronous Boolean
networks via a particular matrix product called the Semi-Tensor Product (STP) of matrices, which
multiplies two matrices and in which the column dimension of the first matrix is not necessarily equal
to the row dimension of the second matrix, but is possibly a multiple or divisor of it. The state space of
a Boolean network of nodes is denoted herein by a vector of states in natural order obtained as the STP
of 2-element vectors representing the network variables. A notable contribution of the paper is that its
matrix expression of logic follows the conventional truth-table order, and not the reverse unfamiliar order
followed so far by the STP community. We reproduce the STP analysis of a classical example network.
We include minute details that make the STP manipulations easily accessible to and more understandable
by their potential users. Our analysis points to more efficient implementations of the STP solution in
which the and binary operations do not inflict a cost more than they really deserve.