Flattening in (Tissue) P Systems

One of the main ideas of membrane systems as introduced by Gheorghe Păun in [7] is the distributed way of computation in the different membrane regions of a membrane system. On the other hand, even for the original variant of membrane systems using catalysts it has been shown that all computations can be carried out in only one single membrane for getting computational completeness (see [3]). Using the idea of flattening which we are going to discuss in this note, especially for P systems working in the sequential transition mode, one often can show that the number of membranes does not matter; for example, as is well known, with transition P systems using only non-cooperative rules one can characterize the family of Parikh sets of regular languages, no matter how many membranes are used. Whereas without any doubt for communication P systems, where computations are carried out by moving objects through membranes, the underlying membrane structure of a P system or the underlying graph structure of a tissue P system will always play an essential role, in the case of transition P systems or tissue P systems with evolution rules a flattening procedure may allow for reducing the number of membranes or cells to one, i.e., to pure multiset rewriting, without changing the main concept for the computational power of such systems. Yet depending on the exact definitions of how these systems are supposed to use their rules and how to get the final results, specific issues have to be discussed carefully. This note addresses to experts in the area of P systems; hence, in general we only refer the reader to [8] and the P page [9] for specific notions and results used or stated afterwards. Formal definitions for a general model of static (tissue) P systems can be found in [5], a formal framework for dynamically evolving structures in [4].