Symbolic dynamics approach to higher order Boolean control networks

The Artin-Mazur Zeta function and topological entropy are two main tools for studying symbolic dynamics. The Artin-Mazur Zeta function includes a number of cycles of different length, and topological entropy reflects the exponential growth rate of the number of n-blocks in symbolic dynamics. Using the semi-tensor product of matrices, under the framework of symbolic dynamics, we investigate the topological structure of higherorder Boolean control networks, which is a special class of higher-order logical control systems. First, a one-to-one correspondence between a k -order logical control network and k -step finite shift of finite type (or k -step finite SFT for short) is constructed. Then, a number of cycles of different length are obtained from the Artin-Mazur Zeta function, and control information from the higher-order Boolean network is learned from the topological entropy. The main contribution of this paper is to provide one-to-one correspondence between a k -order logical control network and k -step finite SFT.