Membrane Division, Oracles, and the Counting Hierarchy.

Polynomial-time P systems with active membranes characterise PSPACE by exploiting membranes nested to a polynomial depth, which may be subject to membrane division rules. When only elementary (leaf) membrane division rules are allowed, the computing power decreases to PPP = P#P, the class of problems solvable in polynomial time by deterministic Turing machines equipped with oracles for counting (or majority) problems. In this paper we investigate a variant of intermediate power, limiting membrane nesting (hence membrane division) to constant depth, and we prove that the resulting P systems can solve all problems in the counting hierarchy CH, which is located between PPP and PSPACE. In particular, for each integer k ≥⃒ 0 we provide a lower bound to the computing power of P systems of depth k. [ABSTRACT FROM AUTHOR]