Subroutines in P systems and closure properties of their complexity classes

The literature on membrane computing describes several variants of P systems whose complexity classes C are “closed under exponentiation”, that is, they satisfy the inclusion P^C subseteq C, where P^C is the class of problems solved by polynomial-time Turing machines with oracles for problems in C. This closure automatically implies closure under many other operations, such as regular operations (union, concatenation, Kleene star), intersection, complement, and polynomial-time mappings, which are inherited from P. Such results are typically proved by showing how elements of a family Pi of P systems can be embedded into P systems simulating Turing machines, which exploit the elements of Pi as subroutines. Here we focus on the latter construction, providing a description that, by abstracting from the technical details which depend on the specific variant of P system, describes a general strategy for proving closure under exponentiation. We also provide an example implementation using polarizationless P systems with active membranes and minimal cooperation.