P systems: Some recent results and research problems

Abstract

Let R = {r1, ...,rk} be the set of labeled rules in a P system. We look at the computing power of the system under three semantics of parallelism. For a positive integer n ≤ k, define: n-Max-Parallel: At each step, nondeterministically select a maximal subset of at most n rules in R to apply. ≤ n-Parallel: At each step, nondeterministically select any subset of at most n rules in R to apply. n-Parallel: At each step, nondeterministically select any subset of exactly n rules in R to apply. Note that in all three cases, at most one instance of any rule can be included in the selected subset. Moreover, if any rule in the subset selected is not applicable, then the whole subset is not applicable. When n = 1, the three semantics reduce to the Sequential mode. For two models of P systems that have been studied in the literature, catalytic systems and communicating P systems, we show that n-Max-Parallel mode is strictly more powerful than any of the following three modes: Sequential, ≤ n-Parallel, or n-Parallel. For example, it follows from a previous result that a 3-Max Parallel communicating P system is universal. However, under the three limited modes of parallelism, the system is equivalent to a vector addition system, which is known to only define a recursive set. This shows that "maximal parallelism" (in the sense of n-Max-Parallel) is key for the model to be universal. We also summarize our recent results concerning membrane hierarchy, determinism versus nondeterminism, and computational complexity of P systems. Finally, we propose some problems for future research. Some of the results presented here were obtained in collaboration with Zhe Dang and Hsu-Chun Yen. © Springer-Verlag Berlin Heidelberg 2005.