P systems with proteins: a new frontier when membrane division disappears

P systems with active membranes are usually defined as devices hierarchically structured that evolve through rewriting rules. These rules take the inspiration on the chemical reactions that happen within a cell and the role of both the inner and the plasma membranes as a filter, letting components pass or not. Classically, these systems are non-cooperative, that is, the left-hand side of the rules has at most one object. Using polarizations, dissolution or cooperation, these systems have been proved to have enough power to efficiently solve computationally hard problems, obtaining new complexity frontiers with respect to their non-cooperative counterparts. In this paper, division rules are interchanged by separation rules. While the first ones produce two new membranes and two new objects, duplicating the objects within the original one, separation rules distribute the objects of the original membrane into the two new created membranes, so no new objects are created in this way. To obtain new objects, a rule of the type [a -> a(2)] would be needed to accomplish that feature that seems to be necessary to obtain efficient solutions to NP-complete problems. Here, we present the limits when using separation rules instead of division rules.