Subobject Transformation Systems and Elementary Net Systems

Graph transformation systems (GTSs) [10] are a powerful specification formalism for concurrent and distributed systems, generalising another classical model of concurrency, namely Place/Transition Petri nets [8]. Along the years, the concurrent behaviour of GTSs has been deeply studied and a consolidated theory of concurrency is now available. In particular, by exploiting the relationship with Petri nets, several concurrent semantics developed for nets, like (deterministic and non-deterministic) processes and the unfolding construction, have been extended to GTSs (see, e.g., [5, 9, 2, 3]). Recently, the algebraic Double-Pushout (DPO) Approach to graph transformation has been extended to rewriting in arbitrary adhesive categories [7, 6], and in cooperation with Paolo Baldan and Barbara König the authors are working on the generalization of the concurrent semantics of nets and GTS to this more abstract setting: first results concerning deterministic processes appeared in [1]. A key ingredient in the definition of processes and unfoldings for a given transformation system, is the analysis of the relationships that emerge among the occurrences of rules in the possible computations of the system. Such relations include the classical parallel and sequential independence, causality and conflict, asymmetric conflict, and the less known cocausality, disabling and codisabling, introduced in [1]. Actually, such relations are meaningful for restricted classes of transformation systems only, including occurrence systems, which satisfy suitable safety and acyclicity conditions. Even if the possible ways in which the rules of an occurrence system can be related are quite well understood, a systematic study of this topic is still missing, and it is an ongoing work by the authors who introduced, to this aim, Subobject Transformation System (STSs) [4]. Intuitively, STSs can be understood as DPO rewriting systems in the category of subobjects of a given object of an adhesive category (corresponding to the type graph in the typed approches to DPO). In this framework, the usual pushout and pullback constructions are replaced by union and intersection of subobjects. In general, one works with a set theoretical syntax rather than with a categorical one. The STS framework is exploited to identify possible basic relations among rules, using them to define other derived relations which are shown to coincide with those used in the literature, and to recast known results about parallel and sequential independence. Next a construction is presented that builds an STS from a given derivation tree of a DPO system, and it is shown that the analysis of