Symbol/Membrane Complexity of P Systems with Symport/Antiport Rules

Abstract

We consider P systems with symport/antiport rules and small numbers of symbols and membranes and present several results for P systems with symport/antiport rules simulating register machines with the number of registers depending on the number s of symbols and the number m of membranes. For instance, any recursively enumerable set of natural numbers can be generated (accepted) by systems with s ≥ 2 symbols and m ≥ 1 membranes such that m + s ≥ 6. In particular, the result of the original paper [17] proving universality for three symbols and four membranes is improved (e.g., three symbols and three membranes are sufficient). The general results that P systems with symport/antiport rules with s symbols and m membranes are able to simulate register machines with max{m(s-2),(m-1)(s-1)} registers also allows us to give upper bounds for the numbers s and m needed to generate/accept any recursively enumerable set of k-dimensional vectors of non-negative integers or to compute any partial recursive function f : ℕ^α →ℕ^β. Finally, we also study the computational power of P systems with symport/antiport rules and only one symbol: with one membrane, we can exactly generate the family of finite sets of non-negative integers; with one symbol and two membranes, we can generate at least all semilinear sets. The most interesting open question is whether P systems with symport/antiport rules and only one symbol can gain computational completeness (even with an arbitrary number of membranes) as it was shown for tissue P systems in [1].