Bounding the space in P systems with active membranes

P systems with active membranes have been widely used to attack problems in NP or even in PSPACE; in general, an exponential amount of space is generated in polynomial time by dividing existing membranes. Natural questions arise in this framework, concerning the power of P systems when different bounds are considered for the use of the space resource. We consider in this paper two natural bounds: the amount of available physical space (in terms of the number of objects and membranes) and the organization of the membrane structure (in particular, concerning the depth of the membrane structure). We present the main results obtained so far on this subject.