Elsevier

Theoretical Computer Science

Volume 324, Issue 1, 16 September 2004, Pages 61-85
Theoretical Computer Science

On the power of membrane division in P systems

https://doi.org/10.1016/j.tcs.2004.03.053Get rights and content
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Abstract

First, we consider P systems with active membranes, hence with the possibility that the membranes can be divided, with non-cooperating evolution rules (the objects always evolve separately). These systems are known to be able to solve NP-complete problems in linear time. Here we give a normal form theorem for such systems: their computational universality is preserved even if only the elementary membranes are divided. The possibility of solving SAT in linear time is preserved only when non-elementary membranes may also be divided under the influence of objects in their region.

Second, we consider a slight generalization, namely, we allow that a membrane can produce by division both a copy of itself and a copy of a membrane with a different label; again, only elementary membranes may be divided. In this case, we prove that the hierarchy on the maximal number of membranes present in the system collapses: three membranes at a time are sufficient in order to characterize the recursively enumerable sets of vectors of natural numbers. This result is optimal, two membranes are shown not to be sufficient.

Third, we consider P systems with cooperating rules (several objects may evolve together). Making use of this powerful feature, we show that many NP-complete problems can be solved in linear time in a quite uniform way (by systems which are very similar to each other), using only elementary membranes division (and not further ingredients, such as electrical charges). The degree of cooperation is minimal: two objects at a time.

Keywords

Membrane computing
Recursively enumerable language
Universality
SAT problem

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Research supported by “Research for Future” Program no. JSPS-RFTF 96I00101, from the Japan Society for the Promotion of Science, and by grant-in-aid for Exploratory Research, No. 11878055 and No. 11837005, from the Ministry of Education, Science, Sports, and Culture of Japan.