A Survey of Elementary Object Systems Part I: Decidability Results

This contribution presents the formalism of Elementary Object Systems (Eos). Object nets are Petri nets which have Petri nets as tokens – an approach known as the nets-within-nets paradigm. Since object nets in general are immediately Turing complete, we introduce the restricted class of elementary object nets which restrict the nesting of nets to the depth of two. In this work we study the relationship of Eos to existing Petri net formalisms. It turns out that Eos are more powerful than classical p/t nets which is demonstrated by the fact that e.g. reachability and liveness become undecidable problems for Eos. Despite these undecidability results other properties can be extended to Eos using a monotonicity argument similar to that for p/t nets. Also linear algebraic techniques, especially the theory of linear invariants and semiflows, can be extended in an appropriate way. The invariant calculus for Eos even enjoys the property of compositionality, i.e. invariants of the whole system can be composed of invariants of the object nets, which reduces the computational effort. To obtain a finer level of insight we also study several subclasses. Among these variants the subclass of generalised state machines is worth mentioning since it combines the decidability of many theoretically interesting properties with a quite rich practical modelling expressiveness. We also study safe Eos, a generalisation of safe p/t nets which are bounded systems with bound b = 1. Four different variants of safeness are studied. It turns out that variants are equivalent for p/t like Eos. While reachability and liveness remain undecidable for the two weaker classes of safe Eos, the two most strongest variants are restrictive enough to guarantee decidability. In fact, both problems are Pspace-complete.