Algebraic approach to graph transformation based on single pushout derivations

Abstract

The Berlin approach to graph transformation, which uses double pushout derivations in the category of graphs and total graph morphisms, is modified using single pushout derivations in the category of graphs and partial graph morphisms. It is shown that the single pushout approach generalizes the classical approach in the sense that all double pushout derivations correspond to single pushout transformations but not vice versa. The chances which lie in the extended expressive power are exhibited in the following. We show that some complex proofs within the framework of double pushout derivations become much simpler in the new context. Moreover, the simple derivation structure allows to consider asynchronous derivations which might provide an adequate model for distributed computations. Finally, the approach is generalized in order to be applicable to more general algebraic structures. We characterize these so-called graph structures as categories of algebras w.r.t. signatures containing unary operator symbols only. Many representations of graphs and hypergraphs known from the literature turn out to be special graph structures such that the theoretical framework introduced in this paper can be applied to all of those objects.