The power of maximal parallelism in P systems

Abstract

We consider the following definition (different from the standard definition in the literature) of "maximal parallelism" in the application of evolution rules in a P system G: Let R = {r1, ⋯rk} be the set of (distinct) rules in the system. G operates in maximal parallel mode if at each step of the computation, a maximal subset of R is applied, and at most one instance of any rule is used at every step (thus at most k rules are applicable at any step). We refer to this system as a maximally parallel system. We look at the computing power of P systems 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 (this implies that no larger subset is applicable). ≤ 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. In all three cases, 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. We focus on two popular models of P systems: multi-membrane catalytic systems and communicating P systems. We show that for these systems, 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 the result in [7] that a maximally parallel communicating P system is universal for n = 2. 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. These generalize and refine the results for the case of 1-membrane systems recently reported in [3]. Some of the present results are rather surprising. For example, we show that a Sequential 1-membrane communicating P. © Springer-Verlag Berlin Heidelberg 2004.