Catalytic P systems, semilinear sets, and vector addition systems

We look at 1-region membrane computing systems which only use rules of the form Ca → Cv, where C is a catalyst, a is a noncatalyst, and v is a (possibly null) string of noncatalysts. There are no rules of the form a → v. Thus, we can think of these systems as "purely" catalytic. We consider two types: (1) when the initial configuration contains only one catalyst, and (2) when the initial configuration contains multiple catalysts. We show that systems of the first type are equivalent to communication-free Petri nets, which are also equivalent to commutative context-free grammars. They define precisely the semilinear sets. This partially answers an open question (in: WMC-CdeA'02, Lecture Notes in Computer Science, vol. 2597, Springer, Berlin, 2003, pp. 400-409; Computationally universal P systems without priorities: two catalysts are sufficient, available at http://psystems.disco.unimib.it, 2003). Systems of the second type define exactly the recursively enumerable sets of tuples (i.e., Turing machine computable). We also study an extended model where the rules are of the form q:(p, Ca → Cv) (where q and p are states), i.e., the application of the rules is guided by a finite-state control. For this generalized model, type (1) as well as type (2) with some restriction correspond to vector addition systems. Finally, we briefly investigate the closure properties of catalytic systems.