Graph transformation systems have been studied extensively and applied to several areas of computer science like formal language theory, the modeling of databases, concurrent or distributed systems, and visual, logical, and functional programming. In most kinds of applications it is necessary to have the possibility of restricting the applicability of rules. This is usually done by means of application conditions. In this paper, we continue the work of extending the fundamental theory of graph transformation to the case where rules may use arbitrary (nested) application conditions. More precisely, we generalize the Embedding theorem, and we study how local confluence can be checked in this context. In particular, we define a new notion of critical pair which allows us to formulate and prove a Local Confluence Theorem for the general case of rules with nested application conditions. All our results are presented, not for a specific class of graphs, but for any arbitrary M-adhesive category, which means that our results apply to most kinds of graphical structures. We demonstrate our theory on the modeling of an elevator control by a typed graph transformation system with positive and negative application conditions.

Lazy graph transformation
(2012)

Applying an attributed graph transformation rule to a given object graph always implies some kind of constraint solving. In many cases, the given constraints are almost trivial to solve. For instance, this is the case when a rule describes a transformation G double right arrow H, where the attributes of H are obtained by some simple computation from the attributes of G. However there are many other cases where the constraints to solve may be not so trivial and, moreover, may have several answers. This is the case, for instance, when the transformation process includes some kind of searching. In the current approaches to attributed graph transformation these constraints must be completely solved when defining the matching of the given transformation rule. This kind of early binding is well-known from other areas of Computer Science to be inadequate. For instance, the solution chosen for the constraints associated to a given transformation step may be not fully adequate, meaning that later, in the search for a better solution, we may need to backtrack this transformation step.
In this paper, based on our previous work on the use of symbolic graphs to deal with different aspects related with attributed graphs, including attributed graph transformation, we present a new approach that, based on the new notion of narrowing graph transformation rule, allows us to delay constraint solving when doing attributed graph transformation, in a way that resembles lazy computation. For this reason, we have called lazy this new kind of transformation. Moreover, we show that the approach is sound and complete with respect to standard attributed graph transformation. A running example, where a graph transformation system describes some basic operations of a travel agency, shows the practical interest of the approach.

Inheritance is an important and widely spread concept enabling the elegant expression of hierarchy in object-oriented software programs or models. It has been defined for graphs and graph transformations enhancing the applicability of this formal technique. Up to now, for the analysis of transformations with inheritance a flattening construction has been used, which yields all the well-known results for graph transformation but results in a large number of graphs and rules that have to be analyzed. In this paper, we introduce a new category of typed attributed graphs with inheritance. For the detection of conflicts between graph transformations on these graphs, the notion of abstract critical pairs is defined. This allows us to perform the analysis on polymorphic rules and transformations without the need for flattening, which significantly increases the efficiency of the analysis and eases the interpretation of the analysis results. The new main result is the Local Confluence Theorem for typed attributed graph transformation with inheritance using abstract critical pairs. All constructions and results are demonstrated on an example for the analysis of refactorings. (C) 2012 Elsevier B.V. All rights reserved.

Graphs are ubiquitous in Computer Science. For this reason, in many areas, it is very important to have the means to express and reason about graph properties. In particular, we want to be able to check automatically if a given graph property is satisfiable. Actually, in most application scenarios it is desirable to be able to explore graphs satisfying the graph property if they exist or even to get a complete and compact overview of the graphs satisfying the graph property.
We show that the tableau-based reasoning method for graph properties as introduced by Lambers and Orejas paves the way for a symbolic model generation algorithm for graph properties. Graph properties are formulated in a dedicated logic making use of graphs and graph morphisms, which is equivalent to firstorder logic on graphs as introduced by Courcelle. Our parallelizable algorithm gradually generates a finite set of so-called symbolic models, where each symbolic model describes a set of finite graphs (i.e., finite models) satisfying the graph property. The set of symbolic models jointly describes all finite models for the graph property (complete) and does not describe any finite graph violating the graph property (sound). Moreover, no symbolic model is already covered by another one (compact). Finally, the algorithm is able to generate from each symbolic model a minimal finite model immediately and allows for an exploration of further finite models. The algorithm is implemented in the new tool AutoGraph.

Graph repair, restoring consistency of a graph, plays a prominent role in several areas of computer science and beyond: For example, in model-driven engineering, the abstract syntax of models is usually encoded using graphs. Flexible edit operations temporarily create inconsistent graphs not representing a valid model, thus requiring graph repair. Similarly, in graph databases—managing the storage and manipulation of graph data—updates may cause that a given database does not satisfy some integrity constraints, requiring also graph repair. We present a logic-based incremental approach to graph repair, generating a sound and complete (upon termination) overview of least-changing repairs. In our context, we formalize consistency by so-called graph conditions being equivalent to first-order logic on graphs. We present two kind of repair algorithms: State-based repair restores consistency independent of the graph update history, whereas deltabased (or incremental) repair takes this history explicitly into account. Technically, our algorithms rely on an existing model generation algorithm for graph conditions implemented in AutoGraph. Moreover, the delta-based approach uses the new concept of satisfaction (ST) trees for encoding if and how a graph satisfies a graph condition. We then demonstrate how to manipulate these STs incrementally with respect to a graph update.

Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church-Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of M-adhesive categories, where M-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules.