A topology for P-systems with active membranes

This paper proposes a study of deterministic P systems with active membranes in the context of discrete time dynamical systems. First of all, we prove that, for a fixed set of objects and labels, the set of all P system configuration is countable and that the dynamical behaviors defining a chaotic system are not possible. Then, we define a notion of distance between membrane configurations encoding the intuitive concept of "dissimilarity" between configurations. We prove that all functions defined by evolution, communication, and division rules are continuous under that distance and that the resulting topological space is discrete but not complete. Furthermore, we adapt in a natural way the classical notions of sensitivity to initial conditions and topological transitivity to P systems, and we show that P systems exhibiting those new properties exist. Finally, we prove that the proposed distance is efficiently computable, i.e., its computation only requires polynomial time with respect to the size of the input configurations.