- Graph rewriting
In
graph theory , graph rewriting is a system ofrewriting for graphs, i.e. a set of graph rewrite rules of the form p: L ightarrow 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 thesubgraph 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 language s; 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
morphism s in the category of graphs with "total"graph morphism s as arrows: r = (L leftarrow K ightarrow R) (or L supseteq K subseteq R) where K ightarrow L isinjective . 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 samemorphism kcolon K ightarrow G (this is where the name "double"-pushout comes from). Anothergraph morphism mcolon L ightarrow 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 (seesubgraph 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 categorylabeled multigraph s with "partial"graph morphism s as arrows: rcolon L ightarrow 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 [http://www.mat2gra.info/ mat2gra.info] .
Yet another approach to graph rewriting, known as "determinate" graph rewriting, came out of
logic anddatabase 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
** [http://tfs.cs.tu-berlin.de/agg/ AGG] , the attributed graph grammar system (Java)
* Tools that solvesoftware engineering tasks (mainly MDA) with graph rewriting:
**GReAT
**VIATRA
** [http://www.fujaba.de/ Fujaba] uses Story driven modelling, a graph rewrite language based on PROGRESReferences
*"Handbook of Graph Grammars and Computing by Graph Transformations". Volume 1-3. World Scientific Publishing
ee also
*
Graph theory
*Category theory
*Graph transformation
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