Symport/Antiport P Systems with Membrane Separation Characterize P^(#P)

TL;DR

This paper characterizes the computational power of symport/antiport membrane systems with membrane separation, showing they exactly solve problems in P^(#P) and demonstrating their capabilities with a concrete implementation.

Contribution

It provides an exact complexity characterization of membrane separation P systems with symport/antiport, including a practical implementation for solving MIDSAT.

Findings

Systems with membrane separation characterize P^(#P) complexity.

Implementation solves MIDSAT, a P^(#P)-complete problem.

Rule length limits affect problem-solving power, from NP to P.

Abstract

Membrane systems represent a computational model that operates in a distributed and parallel manner, inspired by the behavior of biological cells. These systems feature objects that transform within a nested membrane structure. This research concentrates on a specific type of these systems, based on cellular symport/antiport communication of chemicals. Results in the literature show that systems of this type that also allow cell division can solve PSPACE problems. In our study, we investigate systems that use membrane separation instead of cell division, for which only limited results are available. Notably, it has been shown that any problem solvable by such systems in polynomial time falls within the complexity class P^(#P). By implementing a system solving MIDSAT, a P^(#P)-complete problem, we demonstrate that the reverse inclusion is true as well, thus providing an exact characterization of the problem class solvable by P systems with symport/antiport and membrane separation. Moreover, our implementation uses rules of length at most three. With this limit, systems were known to be able to solve NP-complete problems, whereas limiting the rules by length two, they characterize P.