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3 2006

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Theory of Constraints and Application Conditions: From Graphs to High-Level Structures

Ehrig H., Ehrig K., Habel A., Pennemann K-H.

Fundamenta Informaticae

2006

Vol. 74, nr 1

135-166

Graph constraints and application conditions are most important for graph grammars and transformation systems in a large variety of application areas. Although different approaches have been presented in the literature already there is no adequate theory up to now which can be applied to different kinds of graphs and high-level structures. In this paper, we introduce a general notion of graph constraints and application conditions and show under what conditions the basic results can be extended from graph transformation to high-level replacement systems. In fact, we use the new framework of adhesive HLR categories recently introduced as combination of HLR systems and adhesive categories. Our main results are the transformation of graph constraints into right application conditions and the transformation from right to left application conditions in this new framework. The transformations are illustrated by a railroad control system with rail net constraints and application conditions.

Fundamental Theory for Typed Attributed Graphs and Graph Transformation based on Adhesive HLR Categories

Ehrig H., Ehrig K., Prange U., Taentzer G.

Fundamenta Informaticae

2006

Vol. 74, nr 1

31-61

The concept of typed attributed graphs and graph transformation is most significant for modeling and meta modeling in software engineering and visual languages, but up to now there is no adequate theory for this important branch of graph transformation. In this article we give a new formalization of typed attributed graphs, which allows node and edge attribution. The first main result shows that the corresponding category is isomorphic to the category of algebras over a specific kind of attributed graph structure signature. This allows to prove the second main result showing that the category of typed attributed graphs is an instance of ''adhesive HLR categories''. This new concept combines adhesive categories introduced by Lack and Soboci\'nski with the well-known approach of high-level replacement (HLR) systems using a new simplified version of HLR conditions. As a consequence we obtain a rigorous approach to typed attributed graph transformation providing as fundamental results the Local Church-Rosser, Parallelism, Concurrency, Embedding and Extension Theorem and a Local Confluence Theorem known as Critical Pair Lemma in the literature.

Adhesive High-Level Replacement Systems: A New Categorical Framework for Graph Transformation

Ehrig H., Padberg J., Prange U.

Fundamenta Informaticae

2006

Vol. 74, nr 1

1-29

Adhesive high-level replacement (HLR) systems are introduced as a new categorical framework for graph transformation in the double pushout (DPO) approach, which combines the well-known concept of HLR systems with the new concept of adhesive categories introduced by Lack and Sobociński. In this paper we show that most of the HLR properties, which had been introduced to generalize some basic results from the category of graphs to high-level structures, are valid already in adhesive HLR categories. This leads to a smooth categorical theory of HLR systems which can be applied to a large variety of graphs and other visual models. As a main new result in a categorical framework we show the Critical Pair Lemma for the local confluence of transformations. Moreover we present a new version of embeddings and extensions for transformations in our framework of adhesive HLR systems.