Subroutines in P systems and closure properties of their complexity classes

Abstract

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

Simulating counting oracles with cooperation

“…It is quite easy to simulate efficiently (actually, in linear time) a deterministic Turing machine working in polynomial time by means of a uniform family of P systems [8].…”

“…Thus, a single membrane (or a number of membranes obtained by division of a single initial one, in the case of nondeterminism) can efficiently simulate a polynomial-size tape Turing machine and, in particular, a Turing machine working in polynomial time. On the other hand, using several nested membranes it is possible to efficiently simulate oracle queries [8]. With bidirectional P systems (i.e., standard P systems using both send-in and send-out rules) the simulation of the Turing machine is paused, then one usually sends the query string into a child membrane, where another Turing machine for the oracle language is simulated, possibly using membrane division; the answer is then sent out and the simulation of the original Turing machine is resumed.…”

“…With monodirectional P systems we proceed in the opposite direction: we first duplicate and send out the multiset encoding the configuration of the Turing machine being simulated, then the oracle machine is simulated in the innermost membrane, and the result is sent out, where the simulation of the original Turing machine can then resume [8]. This process is depicted in Fig.…”

Cell-like P systems with polarizations and minimal rules

“…The computational complexity of (tissue) P systems with membrane division or membrane separation has also been investigated. In [58], it was proved that P systems with active membranes characterize PSPACE, that is, the class of problems solvable by P systems with active membranes is equal to PSPACE (further studied in [16][17][18][19]57]). In [55,56], an upper bound of the computational power of tissue P systems with cell separation or cell division is provided, respectively; it was thus shown that the class of problems that are solved in polynomial time by tissue P systems with cell separation or cell division is contained in the class PSPACE.…”

P systems attacking hard problems beyond NP: a survey

“…Note the use of oracles and complexity classes of the form , containing languages recognizable by polynomial-time Turing machines with oracles for languages in . Such classes allow for a general way of proving their closure under several operations (union, concatenation, Kleene star, intersection, complement...) implied by their closure under the so-called exponentiation, i.e., ⊆ [27]. The complexity class PSPACE plays an important role in the theory of parallel computing.…”