Zero-Safe Nets: Comparing the Collective and Individual Token Approaches

The main feature of zero-safe nets is a primitive notion of transition synchronization. To this aim, besides ordinary places, called stable places, zero-safe nets are equipped with zero places, which in an observable marking cannot contain any token. This yields the notion of transaction: a basic atomic computation, which may use zero tokens as triggers, but defines an evolution between observable markings only. The abstract counterpart of a generic zero-safe net B consists of an ordinary P/T net whose places are the stable places of B, and whose transitions represent the transactions of B. The two nets offer both the refined and the abstract model of the same system, where the former can be much smaller than the latter, because of the transition synchronization mechanism. Depending on the chosen approach -- collective vs individual token philosophy -- two notions of transaction may be defined, each leading to different operational and abstract models. Their comparison is fully discussed on the basis of a multicasting system example. In the second part of the paper, we make use of category theory to analyze and motivate our framework. More precisely, the two operational semantics of zero-safe nets are characterized as adjunctions, and the derivation of abstract P/T nets as coreflections.