On determinism versus nondeterminism in P systems

An important open problem in the area of membrane computing is whether there is a model of P systems for which the nondeterministic version is strictly more powerful than the deterministic version. We resolve this problem in the following sense--we exhibit two classes of P system acceptors with only communicating rules and show: 1. For the first class, the deterministic and nondeterministic versions are equivalent if and only if deterministic and nondeterministic linear bounded automata are equivalent. The latter problem is a long-standing open question in complexity theory. 2. For the second class, the deterministic version is strictly weaker than the nondeterministic version. Both classes are nonuniversal, but can accept fairly complex languages.