Predictors for flat membrane systems

In this paper we investigate dynamic causalities in membrane systems by proposing the concept of “predictor” originally defined in the context of the reaction systems by Brijder, Ehrenfeucht and Rozenberg. The goal is to characterise sufficient and necessary conditions for the presence of a multiset of molecules of interest in the configuration of a P system at a given evolution step (independently from the non-deterministic choices taken). These conditions can be used to study causal relationships between molecules and, therefore, to predict some aspects of future development of multiset rewriting systems. To achieve this goal, we introduce the new concept of “multiset pattern” representing a logical formula on multisets. A sufficient predictor can be expressed as a pattern characterising initial multisets that will surely evolve, after the given number of evolution steps, into a multiset containing the molecules of interest. On the other hand, a necessary predictor models initial multisets that may evolve after the given number of evolution steps, into a multiset containing the molecules of interest. Necessary predictors can be used to characterise initial multisets that will surely not evolve (in the required number of steps) into a multiset that contains such molecules. We inductively define operators able to compute these predictors. The patterns obtained from our operators are sound (sufficient or necessary) predictors, but, in general, they are not complete.