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# Graph rewriting

In graph theory, graph rewriting is a system of rewriting for graphs, i.e. a set of graph rewrite rules of the form $p: L rightarrow R$, with $L$ being called pattern graph (or left-hand side) and $R$ being called replacement graph (or right-hand side of the rule). A graph rewrite rule is applied to the host graph by searching for an occurrence of the pattern graph (thus solving the subgraph isomorphism problem) and by replacing the found occurrence by an instance of the replacement graph.

Sometimes graph grammar is used as a synonym for graph rewriting system, especially in the context of formal languages; the different wording is used to emphasize the goal of enumerating all graphs from some starting graph, i.e. describing a graph language - instead of transforming a given state (host graph) into a new state.

## Graph rewriting approaches

There are several approaches to graph rewriting, one of them is the algebraic approach, which is based upon category theory. Actually the algebraic approach is divided into some sub approaches, the double-pushout approach (DPO) and the single-pushout approach (SPO) being the most important ones; further on there are the sesqui-pushout and the pullback approach''.

From the perspective of the DPO approach a graph rewriting rule is a pair of morphisms in the category of graphs with total graph morphisms as arrows: $r = \left(L leftarrow K rightarrow R\right)$ (or $L supseteq K subseteq R$) where $K rightarrow L$ is injective. The graph K is called invariant or sometimes the gluing graph. A rewriting step or application of a rule r to a host graph G is defined by two pushout diagrams both originating in the same morphism $kcolon Krightarrow G$ (this is where the name double-pushout comes from). Another graph morphism $mcolon Lrightarrow G$ models an occurrence of L in G and is called a match. Practical understanding of this is that $L$ is a subgraph that is matched from $G$ (see subgraph isomorphism problem), and after a match is found, L is replaced with R in host graph G where K serves as some kind of interface.

In contrast a graph rewriting rule of the SPO approach is a single morphism in the category labeled multigraphs with partial graph morphisms as arrows: $rcolon Lrightarrow R$. Thus a rewriting step is defined by a single pushout diagram. Practical understanding of this is similar to the DPO approach. The difference is, that there is no interface between the host graph G and the graph G' being the result of the rewriting step.

There is also a more algebraic-like approach to graph rewriting, based mainly on Boolean algebra, called matrix graph grammars. This topic is expanded at mat2gra.info

Yet another approach to graph rewriting, known as determinate graph rewriting, came out of logic and database theory. In this approach, graphs are treated as database instances, and rewriting operations as a mechanism for defining queries and views; therefore, all rewriting is required to yield unique results (up to isomorphism), and this is achieved by applying any rewriting rule concurrently throughout the graph, wherever it applies, in such a way that the result is indeed uniquely defined.

## Implementations and applications

Graphs are an expressive, visual and mathematical precise formalism for modelling of objects (entities) linked by relations; objects are represented by nodes and relations between them by edges. Nodes and edges are commonly typed and attributed. Computations are described in this model by changes in the relations between the entities or by attribute changes of the graph elements. They are encoded in graph rewrite/graph transformation rules and executed by graph rewrite systems/graph transformation tools.

• Tools that are application domain neutral:
• GrGen.NET, the graph rewrite generator, a graph transformation tool emitting C#-code or .NET-assemblies
• AGG, the attributed graph grammar system (Java)
• Tools that solve software engineering tasks (mainly MDA) with graph rewriting:
• GReAT
• VIATRA
• Fujaba uses Story driven modelling, a graph rewrite language based on PROGRES

## References

• Handbook of Graph Grammars and Computing by Graph Transformations. Volume 1-3. World Scientific Publishing