The category of typed Graph Grammars and its Adjunctions with Categories of Derivations

Abstract

Motivated by the work which has been done for Petrinets, the paper presents a categorical approach to graph grammars “in the large”. In the large means, that we define categories of graph grammars, graph transition systems, and graph derivation systems which embody the notion “grammar”, “direct derivation”, and “derivation", respectively, as they are defined in the classical algebraic theory. For this purpose we introduce a suitable notion of graph grammar morphism on “typed graph grammars” in analogy to Petri-nets. A typed graph grammar is a grammar for typed graphs which is a slight generalization of the standard case. The main result shows that the three categories are related by left-adjoint functors. We discuss the relationship of our results to similar results obtained in the Petri-net field, and applications to entity/relationship models.