Rolling gears — hidden order in life

This paper proposes a new tool to investigate the structure, dynamics, and control of Boolean networks. Using semi-tensor product and the matrix expression of logical operations, we convert a logical dynamical system (network) into a conventional discrete-time dynamic system. Then most of the analyzing and design tools developed in control theory are applicable to Boolean networks. Using this new approach, we firstly provide formulas for calculating the attractors and their regions of attraction, which completely depict the topological structure of a Boolean network. Secondly, based on input-state structure, we reveal the product structure of circles of a Boolean network. We call this structure the ldquorolling gearsrdquo, which, we believe, can be used to explain many life phenomena. Thirdly, the controllability and observability of Boolean control systems are studied. Several necessary and sufficient conditions are obtained. In one word, this paper provides a mathematical quantity analysis and design method for Boolean networks.