Graphs and Graph Transformations

Director:Schneider, H.
Period:October 1, 2004 - December 31, 2014
Description:

Graphs are often used as an intuitive aid for the clarification of complex matters. Examples of outside computer science include, e.g., chemistry where molecules are modeled in a graphical way. In computer science, data or control flow charts are often used as well as entity relationship charts or Petri-nets to visualize software or hardware architectures. Graph grammars and graph transformations combine ideas from the fields of graph theory, algebra, logic, and category theory, to formally describe changes in graphs.

Category theory is an attractive tool for the description of different structures in a uniform way, e.g., the different models for asynchronous processes: Petri-Nets are based on standard labeled graphs, state charts use hierarchical graphs, parallel logic programming can be interpreted in a graph-theoretical way using so-called jungles, and the actor systems can be visualized as graphs, whose labeling alphabet is a set of term graphs.

Lately, we have concentrated our attention on a theoretical aspect.
Our work on graph transformation is based on notions borrowed from category theory. The so-called double-pushout approach represents a production by two morphisms starting at a common interface graph. One pushout glues the left-hand side of the production into the context, the other does with the right-hand side. Effectively constructing a derivation step, however, requires finding a pushout complement on the left-hand side. Some people consider this disadvantageous. In 1984, Raoult has proposed to model graph rewriting by a single pushout; Loewe has extensively studied this approach, but the discussion was mainly restricted to injective morphisms. Under this assumption, the approaches are equivalent. Some relevant applications such as term graph rewriting, however, lead to non-injective morphisms. We have examined these cases in detail, and we could show that the equivalence also holds for non-injective cases as long as the handle satisfies some reasonable conditions.

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