Concurrent derivations as single pushout graph grammar processes

Algebraic graph transformations visually support intuition, have a strong theoretical basis, and provide a formal, implementation independent basis for the description of discretely evolving computational systems and their formal and tractable analysis. Graph grammar models of concurrent systems (petri nets, actor systems) have inspired corresponding semantics developments. Recently this led to the introduction of partial orders of concurrent derivations (concurrent computations). A concurrent derivation (CDer) abstracts from the (sequential) order of rule applications in the sequential derivation and thus can be considered as a concurrent process. Complementary , a morphism between two concurrent derivations expresses that the rst is a computational approximation of the second. In this paper we newly introduce non-deterministic concurrent derivations (CTrees) as classes of concurrently equivalent sequential derivation trees. Due to the fact that also innnite computations are represented by CTrees, the category of all CTrees of a given graph grammar has a nal object (the concurrent counterpart of the whole sequential tree of the given grammar) which is approximated by all other CTrees. We show that (syntactical) morphisms between two graph grammars induce corresponding adjunction between the corresponding (semantic) categories of CDers and CTrees respectively. The feasibility of a formalism is based essentially on the way in which syntax and semantics are related. Approaches to integrate both syntax and semantics into a uniform framework are therefore particularly attractive and theoretically rich. Two prominent examples for this are Petri net processes 1,10] and term algebras in algebraic speciications 5]. In this paper we contribute to this aim by showing that the semantics of a graph grammar can be described by one graph grammar representing all possible computations that this grammar can perform. Moreover, there is a (semantical) adjunction induced by (syntactical) morphisms. The investigations in this paper follow the single pushout (SPO) approach in 9]. Originally this was formulated for diiernt classes of categories with par