On Categorical Graph Grammars Integrating Structural Transformations and Operations on Labels

Graph-theoretic structures are an obvious means to reason about systems of asynchronous processes. Their dynamic behaviour can be simulated by applying productions of a graph grammar. The present paper is motivated by looking for a formal method that is able to describe the behaviour of systems of processes that share data structures. We generalize the categorical graph-grammar approach by labelling the graphs with elements of a suitable category rather than with those of an alphabet. Thus, operations can be performed on the labels while the graphs are rewritten. After presenting the fundamental definitions and some properties, we demonstrate the usefulness of the approach by modelling some well-known Petri nets as well as a generalized net the places of which are labelled with graphs. Finally, we show that known theoretical techniques are applicable to the generalized framework by exemplary discussing parallel independence of derivation steps.