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In [[computer science]], '''double pushout graph rewriting''' or (DPO graph rewriting) refers to a mathematical framework for [[graph rewriting]]. It was introduced in as one of the first algebraic approaches to graph rewriting in the article "Graph-grammars: An algebraic approach" (1973).<ref name="ehrig73">"Graph-grammars: An algebraic approach", Ehrig, Hartmut and Pfender, Michael and Schneider, Hans-Jürgen, Switching and Automata Theory, 1973. SWAT'08. IEEE Conference Record of 14th Annual Symposium on, pp. 167-180, 1973, IEEE</ref> It has since been generalized to allow rewriting structures which are not graphs, and to handle negative application conditions,<ref name="ehrig2">"Constraints and application conditions: From graphs to high-level structures", Ehrig, Ehrig, Habel and Pennemann, Graph transformations, pp. 287--303, Springer</ref> among other extensions.
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== Definition ==
 
A DPO graph transformation system (or graph grammar) consists of a finite [[Graph (mathematics)|graph]], which is the starting state, and an a finite or countable set of labeled [[Span (category theory)|spans]] in the [[Category (mathematics)|category]] of finite graphs and graph homomorphisms, which serve as derivation rules. The rule spans are generally taken to be composed of [[monomorphism]]s, but the details can vary.<ref name="habel01">"Double-pushout graph transformation revisited", Habel, Annegret and Müller, Jürgen and Plump, Detlef, Mathematical Structures in Computer Science, vol. 11, no. 05., pp. 637--688, 2001, Cambridge University Press</ref>
 
Rewriting is performed in two steps: deletion and addition.
 
After a match from the left hand side to <math>G</math> is fixed, nodes and edges that are not in the right hand side are deleted. The right hand side is then glued in.
 
Gluing graphs is in fact a [[Pushout_(category_theory)|pushout]] construction in the [[category]] of graphs, and the deletion is the same as finding a pushout complement, hence the name.
 
== Uses ==
 
Double pushout graph rewriting allows the specification of graph transformations by specifying a pattern of fixed size and composition to be found and replaced, where part of the pattern can be preserved. The application of a rule is potentially non-deterministic: several distinct matches can be possible. These can be non-overlapping, or share only preserved items, thus showing a kind of [[concurrency]] known as parallel independence,<ref name="corradini">"Concurrent computing: from Petri nets to graph grammars", Corradini, Andrea, ENTCS, vol. 2, pp. 56--70, 1995, Elsevier</ref> or they may be incompatible, in which case either the applications can sometimes be executed sequentially, or one can even preclude the other.
 
It can be used as a language for software design and programming (usually a variant working on richer structures than graphs is chosen). [[Halting problem|Termination]] for DPO graph rewriting is [[undecidable problem|undecidable]] because the [[Post correspondence problem]] can be reduced to it.<ref name="plump1998termination">, "Termination of graph rewriting is undecidable", Detlef Plump, Fundamenta Informaticae, vol. 33, no. 2, pp. 201--209, 1998, IOS Press</ref>
<!-- does an implementation exist which is more than a prototype? for example, GROOVE can be used to simulate DPO rewriting -->
 
DPO graph rewriting can be viewed as a generalization of [[Petri nets]].<ref name="corradini"/>
 
== Generalization ==
Axioms have been sought to describe categories in which DPO rewriting will work. One possibility is the notion of an [[adhesive category]], which also enjoys many closure properties. Related notions are HLR systems, quasi-adhesive categories and <math>\mathcal{M}</math>-adhesive categories, adhesive HLR categories.<ref name="ahlr">Hartmut Ehrig and Annegret Habel and Julia Padberg and Ulrike Prange, "Adhesive high-level replacement categories and systems", 2004, Springer</ref>
 
The concepts of [[adhesive category]] and HLR system are related (an adhesive category with [[coproduct]]s is a HLR system<ref name="Lack">"Adhesive categories", Stephen Lack and Paweł Sobociński, in ''Foundations of software science and computation structures'', pp. 273--288, Springer 2004</ref>).
 
[[Hypergraph]], [[typed graph]] and [[attributed graph]] rewriting,<ref>"Fundamentals of Algebraic Graph Transformation", Hartmut Ehrig, Karsten Ehrig, Ulrike Prange and Gabriele Taentzer</ref> for example, can be handled because they can be cast as adhesive HLR systems.
 
== Notes ==
<references/>
 
[[Category:Graph algorithms]]
[[Category:Graph rewriting]]

Latest revision as of 01:50, 13 March 2014

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