Fig. 1

Determination of the reachability of a three-node Boolean network with given Boolean functions. Based on the logical operations (a) for this network, the corresponding time transition matrix (b) and the time transition diagram (c) can be obtained by semi-tensor product. A signal flow among five integrated states \({e_{8}^{3}}\to {e_{8}^{5}}\to {e_{8}^{7}}\to {e_{8}^{8}}\to {e_{8}^{4}}\to {e_{8}^{3}}\) is formed as a circle. According to Finding 1, it means that all these five integrated states are reachable, which are highlighted in red, while the other three states \({e_{8}^{1}}\), \({e_{8}^{2}}\) and \({e_{8}^{6}}\) are not reachable, which are highlighted in blue