Editing Graph rewriting (section)

==Graph rewriting approaches==
[[File:GraphRewriteExample color.png|thumb|''Top:'' Example graph rewrite rule ([[Program optimization#Assembly level|optimization]] from compiler construction: multiplication with 2 replaced by addition). ''Bottom:'' Application of the rule to optimize "y=x*2" into "y=x+x".]]

===Algebraic approach===
The algebraic approach to graph rewriting is based upon [[category theory]]. The algebraic approach is further divided into sub-approaches, the most common of which are the ''[[Double pushout graph rewriting|double-pushout (DPO) approach]]'' and the ''[[Single pushout graph rewriting|single-pushout (SPO) approach]]''. Other sub-approaches include 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 and [[graph homomorphism]]s between them: <math>r = (L \leftarrow K \rightarrow R)</math>, also written <math>L \supseteq K \subseteq R</math>, where <math>K \rightarrow L</math> 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 (category theory)|pushout]] diagrams both originating in the same [[morphism]] <math>k\colon K\rightarrow D</math>, where D is a ''context graph'' (this is where the name ''double''-pushout comes from). Another graph morphism <math>m\colon L\rightarrow G</math> models an occurrence of L in G and is called a ''[[Pattern matching|match]]''. Practical understanding of this is that <math>L</math> is a subgraph that is matched from <math>G</math> (see [[subgraph isomorphism problem]]), and after a match is found, <math>L</math> is replaced with <math>R</math> in host graph <math>G</math> where <math>K</math> serves as an interface, containing the nodes and edges which are preserved when applying the rule. The graph <math>K</math> 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 <math>G</math>.

In contrast a graph rewriting rule of the SPO approach is a single morphism in the category of [[labeled multigraph]]s and ''partial mappings'' that preserve the multigraph structure: <math>r\colon L\rightarrow R</math>. Thus a rewriting step is defined by a single [[Pushout (category theory)|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.

From the practical perspective, the key distinction between DPO and SPO is how they deal with the deletion of nodes with adjacent edges, in particular, how they avoid that such deletions may leave behind "dangling edges". The DPO approach only deletes a node when the rule specifies the deletion of all adjacent edges as well (this ''dangling condition'' can be checked for a given match), whereas the SPO approach simply disposes the adjacent edges, without requiring an explicit specification.

There is also another algebraic-like approach to graph rewriting, based mainly on Boolean algebra and an algebra of matrices, called ''matrix graph grammars''.<ref>{{harvnb|Perez|2009}} covers this approach in detail.</ref>

===Determinate graph rewriting===
Yet another approach to graph rewriting, known as ''determinate'' graph rewriting, came out of [[logic]] and [[database theory]].<ref>{{cite web |url=https://www.cs.indiana.edu/~vgucht/p24-gyssens.pdf |title=A Graph-Oriented Object Model for Database End-User Interfaces}}</ref>  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.

===Term graph rewriting===
Another approach to graph rewriting is [[term graph]] rewriting, which involves the processing or transformation of term graphs (also known as ''abstract semantic graphs'') by a set of syntactic rewrite rules.

Term graphs are a prominent topic in programming language research since term graph rewriting rules are capable of formally expressing a compiler's [[operational semantics]]. Term graphs are also used as abstract machines capable of modelling chemical and biological computations as well as graphical calculi such as concurrency models. Term graphs can perform [[automated verification]] and logical programming since they are well-suited to representing quantified statements in first order logic. Symbolic programming software is another application for term graphs, which are capable of representing and performing computation with abstract algebraic structures such as groups, fields and rings.

The TERMGRAPH conference<ref>{{cite web |url=http://www.termgraph.org.uk/ |title=TERMGRAPH}}</ref> focuses entirely on research into term graph rewriting and its applications.