Michael Löwe

Algebraic approach to single-pushout graph transformation

Abstract The single-pushout approach to graph transformation interprets a double-pushout transformation rule of the classical algebraic approach which consists of two total graph morphisms as a single partial morphism from the left- to the right-hand side. The notion of a double-pushout diagram for the transformation process can then be substituted by a single-pushout diagram in an appropriate category of partial morphisms. It can be shown that this kind of transformation generalizes the double-pushout framework. Hence, the classical approach can be seen as a special (and very important) case of the new concept. It can be reobtained from the single-pushout approach by imposing an application condition on the redices which formulates the gluing conditions in the new setting. On the other hand, single-pushout transformations are always possible even if the gluing conditions for the redex are violated. The simpler structure of a direct transformation (one pushout diagram instead of two) simplifies many proofs. Hence, the whole theory for double-pushout transformations including sequential composition, parallel composition, and amalgamation can be reformulated and generalized in the new framework. Some constructions provide new effects and properties which are discussed in detail.

Tutorial Introduction to the Algebraic Approach of Graph Grammars Based on Double and Single Pushouts

The gluing construction on which the algebraic notion of a derivation is based operationally provides a simple and intuitive understanding of graph rewriting. Inheriting the powerful toolbox of category theory, its abstract version as a (single resp. double) pushout leads to highly compact and elegant proofs especially for the basic constructions of sequential and parallel independent derivations as well as for concurrent and amalgamated productions respectively.

Algebraic approach to graph transformation based on single pushout derivations

The Berlin approach to graph transformation, which uses double pushout derivations in the category of graphs and total graph morphisms, is modified using single pushout derivations in the category of graphs and partial graph morphisms. It is shown that the single pushout approach generalizes the classical approach in the sense that all double pushout derivations correspond to single pushout transformations but not vice versa.

The category of typed graph grammars and its adjunctions with categories of derivations

Motivated by the work which has been done for Petri-nets, the paper presents a categorical approach to graph grammars “in the large”. In the large means, that we define categories of graph grammars, graph transition systems, and graph derivation systems which embody the notion “grammar”, “direct derivation”, and “derivation”, respectively, as they are defined in the classical algebraic theory. For this purpose we introduce a suitable notion of graph grammar morphism on “typed graph grammars” in analogy to Petri-nets. A typed graph grammar is a grammar for typed graphs which is a slight generalization of the standard case. The main result shows that the three categories are related by left-adjoint functors. We discuss the relationship of our results to similar results obtained in the Petri-net field, and applications to entity/relationship models.

AGG - An Implementation of Algebraic Graph Rewriting

The Agg-system (Algebraic Graph Grammar System) is a prototype implementation of the algebraic approach to graph transformation [Ehr79]. It has been programmed in EIFFEL and runs on SUN workstations under X Window 11.5. It consists of a flexible graphical editor and a derivation component. The editor allows the graphical manipulation of rules, redices and derivation results. The derivation component performs direct transformation steps for user-selected rules and redices.

Parallel and distributed derivations in the single-pushout approach

Abstract Parallel and distributed derivations are introduced and studied in the single-pushout approach, which models rewriting by pushout constructions in appropriate categories of partial morphisms. We present a categorical framework for this approach in an axiomatic way. Models of this categorical framework are among others: graphs, hypergraphs, relational structures, and algebraic specifications with suitable partial morphisms. Several new results concerning parallelism and distributed parallelism are presented which are even new in the example categories.

Abstract Graph Derivations in the Double Pushout Approach

In the algebraic theory of graph grammars, it is common practice to present some notions or results “up to isomorphism”. This allows one to reason about graphs and graph derivations without worrying about representation-dependent details.

Graph rewriting in span-categories

There are three variations of algebraic graph rewriting, the double-pushout, the single-pushout, and the sesqui-pushout approach. In this paper, we show that all three approaches can be considered special cases of a general rewriting framework in suitable categories of spans over a graph-like base category. From this new view point, it is possible to provide a general and unifying theory for all approaches. We demonstrate this fact by the investigation of general parallel independence. Besides this, the new and more general framework offers completely new ways of rewriting: Using spans as matches, for example, provides a simple mechanism for universal quantification. The general theory, however, applies to these new types of rewriting as well.

Categorical principles, techniques and results for high-level-replacement systems in computer science

The aim of this paper is to give an introduction how to use categorical methods in a specific field of computer science: The field of high-level-replacement systems has its roots in the well-established theories of formal languages, term rewriting, Petri nets, and graph grammars playing a fundamental role in computer science. More precisely, it is a generalization of the algebraic approach to graph grammars which is based on gluing constructions for graphs defined as pushouts in the category of graphs. The categorical theory of high-level-replacement systems is suitable for the dynamic handling of a large variety of high-level structures in computer science including different kinds of graphs and algebraic specifications. In this paper we discuss the basic principles and techniques from category theory applied in the field of high-level-replacement systems and present some basic results together with the corresponding categorical proof techniques.

An Event Structure Semantics for Graph Grammars with Parallel Productions

We propose a truly concurrent semantics for graph grammars, based on event structures, that generalizes to arbitrary consuming grammars (i.e., such that each production deletes some items) the semantics presented in [4] for the subclass of safe grammars. Also, parallel derivations are explicitly considered, instead of sequential ones only as in [4]. The “domain” and the “event structure” of a grammar are introduced independently, and one main result shows that they are strongly related, since the domain is the domain of finite configurations of the event structure. Another important result provides an abstract characterization of when two (parallel) graph derivations should be considered as equivalent from a true-concurrency perspective.