Space complexity equivalence of P systems with active membranes and Turing machines

We prove that arbitrary single-tape Turing machines can be simulated by uniform families of P systems with active membranes with a cubic slowdown and quadratic space overhead. This result is the culmination of a series of previous partial results, finally establishing the equivalence (up to a polynomial) of many space complexity classes defined in terms of P systems and Turing machines. The equivalence we obtained also allows a number of classic computational complexity theorems, such as Savitch's theorem and the space hierarchy theorem, to be directly translated into statements about membrane systems.