Shortest Trajectories and Reversibility in Boolean Automata Networks

§1 Intuition and motivation. It seems that often, when a Boolean Automata Network (BAN) N can make a global change, it can do it rather quickly, i.e. with few local changes. More precisely, when it is possible to reach a certain specific configuration y ∈ Bn = {0, 1} of N , starting from an initial configuration x ∈ Bn of N , then it seams that the following is often the case. To make the global change x y, only a small number of local changes need to be made, i.e. only a small number of automata need to ’move’ (change states) : something (polynomially) comparable to the size n ∈ N of the network N i.e. to the number n of automata in N and to the total number of different conceivable automata moves away from x. To check this conjecture and specify the meaning of “often” in this context, we take interest here in “long trajectories”.