On the computability power of membrane systems with controlled mobility

Abstract

In a previous paper we have shown that membrane systems with controlled mobility are able to solve a Π 2P complete problem. Then, an enriched model with forced endocytosis and forced exocytosis enables us to move to the fourth level in the polynomial hierarchy, the model having Σ 4P ∪ Π 4P as lower bound. In this paper we study the computability power of this model (using forced endocytosis and forced exocytosis), and determine the border condition for achieving computational completeness: 4 membranes provide Turing completeness, while 3 membranes do not. Moreover, we show that the restricted division operation (which is crucial in achieving the Σ 4P ∪ Π 4P lower bound) does not provide computational completeness. However, Turing completeness can be achieved with pairs of operations (exocytosis, inhibitive endocytosis) and (inhibitive exocytosis, endocytosis) by using 4 membranes. Finally, we present some computability results expressing that membrane systems which use the operations of restricted division, restricted exocytosis and inhibitive endocytosis cannot yield computational completeness. © 2012 Springer-Verlag.