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By Gabriel Ciobanu

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For systems with only one membrane the minimal parallelism is nothing else than non-synchronization, hence the non-trivial case is that of multi-membrane systems. We consider only two cases, that of P systems with symport/antiport rules, and of P systems with active membranes. Somewhat surprisingly, the universality is obtained again, in both cases. For instance, for symport/antiport systems, we again need a small number of membranes (three in the generative case, and two in the accepting case), while the symport and antiport rules are rather simple (of weight two).

Of course, as usual for P systems with active membranes, each membrane and each object can be involved in only one rule, and the choice of rules to use and of objects and membranes to evolve is done in a non-deterministic way. We should note that for rules of type (a) the membrane is not considered to be involved: when applying [ h a → v] h , the object a cannot be used by other rules, but the membrane h can be used by any number of rules of type (a) as well as by one rule of types (b) – (e). In each step, the use of rules is done in the bottom-up manner (first the inner objects and membranes evolve, and the result is duplicated if any surrounding membrane is divided).

A system of multisets of rules R is called valid, maximally valid or reversely valid in the skin membrane M if each Ri is valid, maximally valid or reversely valid in membrane Mi , which is the descendant of M with label i, i ∈ {1, . . , m}. 1: Π = (O, µ, w1 , . . wm , R1Π , . . , Rm ) where (u → v) ∈ e R1Π if and only if (v → u) ∈ R1Π . Note that Π = Π. If R = (R1 , . . , Rm ) is a system of multisets of rules for a P system Π, we denote by R the system of multisets of rules for the reverse P system Π given by R = (R1 , .

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