Computations in Graph Rewriting: Inductive Types and Pullbacks in DPO Approach

Graph grammars have been introduced in the late 1970s [1], then they have been significantly improved up to the 2000s [2]. A lot of significant results are due to H. Ehrig and his colleagues who have conceived an algebraic approach to graph rewriting by the means of category theory [1]. It opened the way to computations with attributes. In this approach, when dealing with model transformations, the transformation process can be viewed as split into two parts: a first one considers the skeleton of the models, i.e., graphs without attributes which can be processed by a double pushout in a graph category, and a second one devoted to computations with attributes. To deal with this part, Ehrig suggests another formalism : the theory of algebraic data types [3]. Our goal when designing the Double Pushout-Pullback approach (abbreviated DPOPB) considered in this paper was on one hand to take advantage of the double pushout approach to implement the rewriting of the structural part of the graphs, and on the other hand to unify in a single formalism (type theory) the attribute computations that occur in graph transformations. Generally, to remain in a unique formalism simplifies the implementations and leads to a more robust software. Moreover, we had in mind to furnish a formalism able to facilitate proofs of properties occurring during transformations such as invariant or preor postconditions preservation. Thus, the main idea of the DPoPb approach is the use of a single formalism for attributed graph rewriting. The power of computations with inductive types is greater due to the presence of functional arguments. The formalism also permits to carry on proofs on transformations.