The Joys of Graph Transformation

We believe that the technique of graph transformation offers a very natural way to specify semantics for languages that have dynamic allocation and linking structure; for instance, object-oriented programming languages, but also languages for mobility. In this note we expose, on a rather informal level, the reasons for this belief. Our hope in doing this is to raise interest in this technique and so generate more interest in the fascinating possibilities and open questions of this area. 1 Graph Transformation Is Easy Transformation means changing (literally: shaping) one thing into another. In the case of graph transformation, obviously, the things being changed are graphs. A fundamental assumption in studying such changes is that they are not arbitrary but controlled by some guiding principles, and that these principles can be captured in rules. A graph transformation rule (often called a production rule) describes a kind of change that will transform certain graphs — those to which the rule is applicable — into others, in a specific way encoded in the rule. A set of production rules is usually called a production system; graphs that are subjected to transformation are often called host graphs. For us, the interest in this arises from the fact that graphs can be used to model just about any discrete structure — with lesser or greater ease — and that many kinds of dynamic changes in such structures lend themselves quite naturally to a description by graph production rules. This is in particular true of the semantics of object-oriented systems: it is our firm belief that graph transformations are a very natural technique to specify the semantics of such systems (see also Sect. 4). (It should be mentioned that this is not the only, and indeed historically not the original, reason to be interested in graph transformation: another motivation is to characterise graph structures as the end product of a sequence of transformations guided by a given set of rules. As a very simple example, the class of all connected graphs can be characterised in this way. In that context it is not so much the process of change as its result that is of primary interest.) Although there are many ways to define graph production rules, all rules have certain basic things in common. They always specify changes in a (relatively) small sub-structure, and the change always consists of modifications to that sub-structure, such as taking away parts from it or adding parts to it. For the rule to apply, the first requirement is that the host graph actually contains a sub-structure of the right kind; in fact, if it contains more than one such sub-structure, the rule is applicable in different ways. This is, of course, a very general description; in practice, there have turned out to be many different useful ways to specify sub-structures and changes. In the past this has led to strong opinions about the relative merits of the various techniques. Fortunately, the purpose of this paper is not to categorise these approaches (for that, the handbook [21] is a