Fundamental Theory for Typed Attributed Graph Transformation

The concept of typed attributed 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 paper 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 Sobociński 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.