Catalytic P systems and purely catalytic P systems are among the first variants of membrane systems ever considered in this area. These variants of systems also feature some prominent computational complexity questions, and in particular the problem if only one catalyst in catalytic P systems and two catalysts in purely catalytic P systems are enough to allow for generating all recursively enumerable sets of multisets. Several additional ingredients have been shown to be sufficient for obtaining such results. Previously, we could show that using the derivation mode maxobjects, where we only take those multisets of rules which affect the maximal number of objects in the underlying configuration, one catalyst is sufficient for obtaining computational completeness without any other ingredients in catalytic P systems. In this paper we investigate the question whether we can obtain a similar result for purely catalytic P systems, i.e., we show that two catalysts in purely catalytic P systems are enough to allow for generating all recursively enumerable sets of multisets when using specific variants of the maximally parallel derivation mode: we take only those applicable multisets of rules which (i) generate the maximal number of objects, or (ii) yield the maximal difference in the number of objects between the newly generated configuration and the current configuration. In addition, we also consider non-extendable multisets of rules which (i) generate the minimal number of objects, or (ii) yield the minimal difference in the number of objects between the newly generated configuration and the current configuration. In all cases, we have also shown that register machines with n decrementable registers can be simulated by simple purely catalytic P systems working in any of these derivation modes using only n catalysts. Hence, simple purely catalytic P systems working in any of these derivation modes are computationally complete.

We extend and refine previous results within the general framework for regulated rewriting based on the applicability of rules in sequential grammars [3]. Besides the well-known control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, and priority relations on rules we also consider the new variant of activation and blocking of rules as investigated in [1, 2, 4]. Moreover, we exhibit special results for strings and multisets as well as for arrays in the general variant defined on Cayley grids of finitely presented groups. Especially we prove that array grammars defined on Cayley grids of finitely presented groups using #-context-free array productions together with control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, priority relations on rules, or activation and blocking of rules have the same computational power as such array grammars using arbitrary array productions.

We consider extended spiking neural P systems with the additional possibility of so-called "white hole rules", which send the complete contents of a neuron to other neurons, and we prove that this extension of the original model can easily simulate register machines. Based on this proof, we then define red-green variants of these extended spiking neural P systems with white hole rules and show how to go beyond Turing with these red-green systems. We also discuss the number of actor neurons needed, and the relation of this model to some special variants of Lindenmayer systems.

The questions whether catalytic P systems with only one catalyst and purely catalytic P systems with only two catalysts can already be computationally complete in the generative case, still are open problems. For accepting P systems or P automata, th

We consider a new variant of the halting condition in P systems, i.e., a computation in a P system is already called halting if not for all membranes a rule is applicable anymore at the same time, whereas usually a computation is called halting if no rule is applicable anymore in the whole system. This new variant of partial halting is especially investigated for several variants of P systems using membrane rules with permitting contexts and working in different transition modes, especially for minimal parallelism. Both partial halting and minimal parallelism are based on an arbitrary set of subsets from the set of rules assigned to the membranes.