Graph Rewriting - Graph Rewriting Approaches

Graph Rewriting Approaches

There are several approaches to graph rewriting. One of them is the algebraic approach, which is based upon category theory. The algebraic approach is divided into some sub approaches, the double-pushout approach (DPO) and the single-pushout approach (SPO) being the most common 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: (or ) where 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 (this is where the name double-pushout comes from). Another graph morphism models an occurrence of L in G and is called a match. Practical understanding of this is that is a subgraph that is matched from (see subgraph isomorphism problem), and after a match is found, is replaced with in host graph where serves as an interface, containing the nodes and edges which are preserved when applying the rule. The graph is needed to attach the pattern being matched to its context: if it is empty, the match can only designate a whole connected component of the graph .

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: . 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 another algebraic-like approach to graph rewriting, based mainly on Boolean algebra and an algebra of matrices, called matrix graph grammars.

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.