Graph transformation is a specification technique suitable for a wide range of applications, specially the ones that require a sophisticated notion of state. In graph transformation, states are represented by graphs and actions are specified by rules. Most algebraic approaches to graph transformation proposed in the literature ensure that if an item is preserved by a rule, so are its connections with the graph where it is embedded. But there are applications in which it is desirable to specify different embeddings. For example when cloning an item, there may be a need to handle the original and the copy in different ways. We propose a new algebraic approach to graph transformation, AGREE: Algebraic Graph Rewriting with controllEd Embedding, where rules allow one to specify how the embedding should be carried out. We define this approach in the framework of classified categories which are categories endowed with partial map classifiers. This new approach leads to graph transformations in which effects may be non-local, e.g. a rewrite step may alter a node of the host graph which is outside the image of the left-hand side of the considered rule. We propose a syntactic condition on AGREE rules which guarantees the locality of transformations. We also compare AGREE with other algebraic approaches to graph transformation.

Keywords: Rewrite Systems, Graph Transformation, Algebraic Methods

We propose a new algebraic approach to graph transformation, called the Pullback-Pushout (pbpo ) approach, where we combine smoothly the classical modifications to a host graph specified by a first part of a rule, defined as a span of graph morphisms, with the cloning of structures specified by a second span. The motivation behind this new approach is to support cloning of structures in an elegant and efficient way. After a formal definition of the proposed approach, we demonstrate that pbpo rewriting is a conservative extension of agree and the Sesqui-Pushout approaches. Contrary to agree, we show that the proposed pbpo transformation can easily be extended to cope with attributed graphs. In general, totally attributed graphs are not closed under pbpo transformation. We propose sufficient conditions which guarantee that the attribution of transformed graphs is total. Furthermore, a pbpo transformation can affect all parts of a host graph including non local parts (i.e., parts which are outside the image of the left-hand side of a rule). We propose and discuss some conditions which ensure a form of locality of pbpo transformations.

Keywords: Rewrite systems, Graph transformation, Cloning, Algebraic methods

We investigate garbage collection of unreachable parts of rooted graphs from a categorical point of view. First, we define this task as the right adjoint of an inclusion functor. We also show that garbage collection may be stated via a left adjoint, hence preserving colimits, followed by two right adjoints. These three adjoints cope well with the different phases of a traditional garbage collector. Consequently, our results should naturally help to better formulate graph transformation steps in order to get rid of garbage (unwanted nodes). We illustrate this point on a particular class of graph rewriting systems based on a double pushout approach and featuring edge redirection. Our approach gives a neat rewriting step akin to the one on terms, where garbage never appears in the reduced term.

Dependency analysis consists in finding how different parts of a program depend one from each other. It is the root of many program analyses such as dead-code, binding time, strictness, program slicing etc. We address the problem of dependency analysis in the context of typed λ-calculus. We consider all systems of the λ-cube and extend them conservatively in order to determine which parts of a λ-term may contribute to its evaluation. We show how typing information can be used to statically identify dependencies.