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 PC C, where PC 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 of P systems can be embedded into P systems simulating Turing machines, which exploit the elements of as subroutines. Here we focus on the latter construction, abstracting from the technical details which depend on the speci c variant of P system, in order to describe a general strategy for proving closure under exponentiation.