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Graph rewriting
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<h2 id="graph-rewriting-approaches">Graph rewriting approaches</h2> <h3>Algebraic approach</h3> <p>The algebraic approach to graph rewriting is based upon <a href="/facts/Category_theory/6zS3DXPa">category theory</a>. The algebraic approach is further divided into sub-approaches, the most common of which are the <i><a href="/facts/Double_pushout_graph_rewriting/ILtbNYHX">double-pushout (DPO) approach</a></i> and the <i><a href="/facts/Single_pushout_graph_rewriting/b4lpxA1T">single-pushout (SPO) approach</a></i>. Other sub-approaches include the <i>sesqui-pushout</i> and the <i>pullback approach</i>. </p><p>From the perspective of the DPO approach a graph rewriting rule is a pair of <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> in the category of graphs and <a href="/facts/Graph_homomorphism/XSljQ25z">graph homomorphisms</a> between them: r = ( L ← K → R ) {\displaystyle r=(L\leftarrow K\rightarrow R)} , also written L ⊇ K ⊆ R {\displaystyle L\supseteq K\subseteq R} , where K → L {\displaystyle K\rightarrow L} is <a href="/facts/Injective/5ES6tqf5">injective</a>. The graph K is called <i>invariant</i> or sometimes the <i>gluing graph</i>. A <i><a href="/facts/Rewriting/9PHlnnnJ">rewriting</a> step</i> or <i>application</i> of a rule r to a <i>host graph</i> G is defined by two <a href="/facts/Pushout_(category_theory)/6wuns0Yo">pushout</a> diagrams both originating in the same <a href="/facts/Morphism/Kdpuyz3S">morphism</a> k : K → D {\displaystyle k\colon K\rightarrow D} , where D is a <i>context graph</i> (this is where the name <i>double</i>-pushout comes from). Another graph morphism m : L → G {\displaystyle m\colon L\rightarrow G} models an occurrence of L in G and is called a <i><a href="/facts/Pattern_matching/CEwptMs9">match</a></i>. Practical understanding of this is that L {\displaystyle L} is a subgraph that is matched from G {\displaystyle G} (see <a href="/facts/Subgraph_isomorphism_problem/86LLWRg4">subgraph isomorphism problem</a>), and after a match is found, L {\displaystyle L} is replaced with R {\displaystyle R} in host graph G {\displaystyle G} where K {\displaystyle K} serves as an interface, containing the nodes and edges which are preserved when applying the rule. The graph K {\displaystyle K} 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 G {\displaystyle G} . </p><p>In contrast a graph rewriting rule of the SPO approach is a single morphism in the category of <a href="/facts/Labeled_multigraph/udmS0Cp0">labeled multigraphs</a> and <i>partial mappings</i> that preserve the multigraph structure: r : L → R {\displaystyle r\colon L\rightarrow R} . Thus a rewriting step is defined by a single <a href="/facts/Pushout_(category_theory)/6wuns0Yo">pushout</a> 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. </p><p>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 <i>dangling condition</i> can be checked for a given match), whereas the SPO approach simply disposes the adjacent edges, without requiring an explicit specification. </p><p>There is also another algebraic-like approach to graph rewriting, based mainly on Boolean algebra and an algebra of matrices, called <i>matrix graph grammars</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h3>Determinate graph rewriting</h3> <p>Yet another approach to graph rewriting, known as <i>determinate</i> graph rewriting, came out of <a href="/facts/Logic/8JnpJLtt">logic</a> and <a href="/facts/Database_theory/unyQa3cK">database theory</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> 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 (<a href="/facts/Up_to_isomorphism/ydz4ztzX">up to isomorphism</a>), 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. </p> <h3>Term graph rewriting</h3> <p>Another approach to graph rewriting is <a href="/facts/Term_graph/TXGPqGQV">term graph</a> rewriting, which involves the processing or transformation of term graphs (also known as <i>abstract semantic graphs</i>) by a set of syntactic rewrite rules. </p><p>Term graphs are a prominent topic in programming language research since term graph rewriting rules are capable of formally expressing a compiler's <a href="/facts/Operational_semantics/G24LXz6e">operational semantics</a>. 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 <a href="/facts/Automated_verification/BYD4GnJR">automated verification</a> 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. </p><p>The TERMGRAPH conference<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> focuses entirely on research into term graph rewriting and its applications. </p> <h2 id="classes-of-graph-grammar-and-graph-rewriting-system">Classes of graph grammar and graph rewriting system</h2> <p>Graph rewriting systems naturally group into classes according to the kind of representation of graphs that are used and how the rewrites are expressed. The term graph grammar, otherwise equivalent to graph rewriting system or graph replacement system, is most often used in classifications. Some common types are: </p> <ul><li><a href="/facts/Attributed_graph_grammar/GExzKSKK">Attributed graph grammars</a>, typically formalised using either the <a href="/facts/Single-pushout_approach/b4lpxA1T">single-pushout approach</a> or the <a href="/facts/Double-pushout_approach/ILtbNYHX">double-pushout approach</a> to characterising replacements, mentioned in the above section on the algebraic approach to graph rewriting.</li> <li>Hypergraph grammars, including as more restrictive subclasses port graph grammars, <a href="/facts/Linear_graph_grammar/Jz8PM4rU">linear graph grammars</a> and <a href="/facts/Interaction_net/hdc9aAbT">interaction nets</a>.</li></ul> <h2 id="implementations-and-applications">Implementations and applications</h2> <p>Graphs are an expressive, visual and mathematically 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. </p> <ul><li>Tools that are application domain neutral: <ul><li><a href="https://web.archive.org/web/20110217171142/http://user.cs.tu-berlin.de/~gragra/agg/">AGG</a>, the attributed graph grammar system (<a href="/facts/Java_(programming_language)/9ScgFyAL">Java</a>).</li> <li><a href="https://uoycs-plasma.github.io/GP2/">GP 2</a> is a visual rule-based graph programming language designed to facilitate formal reasoning over graph programs.</li> <li><a href="http://homepages.laas.fr/khalil/GMTE/">GMTE</a> <a href="https://web.archive.org/web/20180313184934/http://homepages.laas.fr/khalil/GMTE/">Archived</a> 2018-03-13 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>, the Graph Matching and Transformation Engine for <a href="/facts/Graph_matching/BlTqeDGP">graph matching</a> and transformation. It is an implementation of an extension of Messmer’s algorithm using <a href="/facts/C%2B%2B/Et0F2qn9">C++</a>.</li> <li><a href="/facts/GrGen/MPwG7IgD">GrGen.NET</a>, the graph rewrite generator, a graph transformation tool emitting <a href="/facts/C_Sharp_(programming_language)/UC2gxeyb">C#</a>-code or .NET-assemblies.</li> <li><a href="http://groove.cs.utwente.nl/">GROOVE</a>, a Java-based tool set for editing graphs and graph transformation rules, exploring the state spaces of graph grammars, and model checking those state spaces; can also be used as a graph transformation engine.</li> <li><a href="https://github.com/Verites/verigraph/">Verigraph</a>, a software specification and verification system based on graph rewriting (<a href="/facts/Haskell_(programming_language)/4Htv9WLu">Haskell</a>).</li></ul></li> <li>Tools that solve <a href="/facts/Software_engineering/KX6cE9kz">software engineering</a> tasks (mainly <a href="/facts/Model-driven_architecture/IY0daKp1">MDA</a>) with graph rewriting: <ul><li><a href="http://emoflon.org/">eMoflon</a>, an EMF-compliant model-transformation tool with support for <a href="/facts/Story-driven_modeling/HCD8Dmbp">Story-Driven Modeling</a> and Triple Graph Grammars.</li> <li><a href="http://www.emorf.org">EMorF</a> <a href="https://web.archive.org/web/20160422081445/http://www.emorf.org/">Archived</a> 2016-04-22 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> a graph rewriting system based on <a href="/facts/Eclipse_Modeling_Framework/Hd939zYh">EMF</a>, supporting in-place and model-to-model <a href="/facts/Model_transformation/M0jdCTM5">transformation</a>.</li> <li><a href="http://www.fujaba.de/">Fujaba</a> uses Story driven modelling, a graph rewrite language based on PROGRES.</li> <li><a href="/facts/Graph_Database/WJfPlV3z">Graph databases</a> often support dynamic rewriting of graphs.</li> <li><a href="/facts/GReAT/As1wyc9W">GReAT</a>.</li> <li><a href="https://tinkerpop.apache.org/gremlin.html">Gremlin</a>, a graph-based programming language (see <a href="https://github.com/tinkerpop/gremlin/wiki/Graph-Rewriting">Graph Rewriting</a>).</li> <li><a href="https://www.eclipse.org/henshin/">Henshin</a>, a graph rewriting system based on <a href="/facts/Eclipse_Modeling_Framework/Hd939zYh">EMF</a>, supporting in-place and model-to-model <a href="/facts/Model_transformation/M0jdCTM5">transformation</a>, <a href="/facts/Critical_pair_(logic)/paPprn9L">critical pair analysis</a>, and <a href="/facts/Model_checking/FwvKMlw9">model checking</a>.</li> <li><a href="http://www.se.rwth-aachen.de/tikiwiki/tiki-index.php%3Fpage=Research%3A+Progres.html">PROGRES</a>, an integrated environment and very high level language for PROgrammed Graph REwriting Systems.</li> <li><a href="/facts/VIATRA/K0FSegBH">VIATRA</a>.</li></ul></li> <li>Mechanical engineering tools <ul><li><a href="http://www.graphsynth.com">GraphSynth</a> is an interpreter and UI environment for creating unrestricted graph grammars as well as testing and searching the resultant language variant. It saves graphs and graph grammar rules as <a href="/facts/XML/clNgUfWk">XML</a> files and is written in <a href="/facts/C_Sharp_(programming_language)/UC2gxeyb">C#</a>.</li> <li><a href="https://archive.today/20150317152250/https://www.soley-technology.com/en/pr-soley-studio">Soley Studio</a>, is an <a href="/facts/Integrated_development_environment/jbEINSQf">integrated development environment</a> for graph transformation systems. Its main application focus is data analytics in the field of engineering.</li></ul></li> <li>Biology applications <ul><li><a href="https://grogra.de/">Functional-structural plant modeling with a graph grammar based language</a></li> <li><a href="https://dx.doi.org/10.1016/j.tcs.2011.07.004">Multicellular development modeling with string-regulated graph grammars</a></li> <li><a href="https://kappalanguage.org/">Kappa</a> is a rule-based language for modeling systems of interacting agents, primarily motivated by molecular systems biology.</li></ul></li> <li>Artificial Intelligence/Natural Language Processing <ul><li><a href="/facts/OpenCog/HS2MMBp3">OpenCog</a> provides a basic pattern matcher (on <a href="/facts/Hypergraph/qIR2o2I3">hypergraphs</a>) which is used to implement various AI algorithms.</li> <li><a href="http://wiki.opencog.org/w/RelEx">RelEx</a> is an English-language parser that employs graph re-writing to convert a <a href="/facts/Link_grammar/2cSELk8L">link parse</a> into a <a href="/facts/Dependency_grammar/cWnwcCM0">dependency parse</a>.</li></ul></li> <li>Computer programming language <ul><li>The <a href="/facts/Clean_programming_language/vvxRMVyz">Clean programming language</a> is implemented using graph rewriting.</li></ul></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Graph_theory/VVPCd4TX">Graph theory</a></li> <li><a href="/facts/Shape_grammar/2mxL0uHH">Shape grammar</a></li> <li><a href="/facts/Formal_grammar/QLxaQNAz">Formal grammar</a></li> <li><a href="/facts/Abstract_rewriting/FNIYHu2k">Abstract rewriting</a> — a generalization of graph rewriting</li></ul> <h3>Citations</h3> <h3>Sources</h3> <ul><li>Rozenberg, Grzegorz (1997), <a href="https://web.archive.org/web/20131004120515/http://www.informatik.uni-trier.de/~ley/db/conf/gg/handbook1997.html"><i>Handbook of Graph Grammars and Computing by Graph Transformations</i></a>, vol. 1–3, World Scientific Publishing, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9810228848, archived from <a href="http://www.informatik.uni-trier.de/~ley/db/conf/gg/handbook1997.html">the original</a> on 2013-10-04, retrieved 2012-07-11.</li> <li>Perez, P.P. (2009), <i>Matrix Graph Grammars: An Algebraic Approach to Graph Dynamics</i>, <a href="/facts/VDM_Verlag/BMVzrjy4">VDM Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-639-21255-6.</li> <li>Heckel, R. (2006). <i>Graph transformation in a nutshell</i>. <a href="http://www.elsevier.com/locate/entcs">Electronic Notes in Theoretical Computer Science</a> 148 (1 SPEC. ISS.), pp. 187–198.</li> <li>König, Barbara (2004). <i>Analysis and Verification of Systems with Dynamically Evolving Structure</i>. <a href="http://www.fmi.uni-stuttgart.de/szs/publications/koenigba/habilschrift.pdf">Habilitation thesis, Universität Stuttgart</a> <a href="https://web.archive.org/web/20070625130924/http://www.fmi.uni-stuttgart.de/szs/publications/koenigba/habilschrift.pdf">Archived</a> 2007-06-25 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>, pp. 65–180.</li> <li>Lobo, Daniel; Vico, Francisco J.; Dassow, Jürgen (2011-10-01). <a href="https://doi.org/10.1016%2Fj.tcs.2011.07.004">"Graph grammars with string-regulated rewriting"</a>. <i>Theoretical Computer Science</i>. 412 (43): 6101–6111. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.tcs.2011.07.004">10.1016/j.tcs.2011.07.004</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10630%2F6716">10630/6716</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0304-3975">0304-3975</a>.</li> <li><a href="/facts/Grzegorz_Rozenberg/48Gc0agC">Grzegorz Rozenberg</a>, ed. (Feb 1997). <a href="https://www.worldscientific.com/worldscibooks/10.1142/3303"><i>Foundations</i></a>. Handbook of Graph Grammars and Computing by Graph Transformation. Vol. 1. World Scientific. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F3303">10.1142/3303</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-2884-2.</li> <li><a href="/facts/Hartmut_Ehrig/Ljv8U1RM">Hartmut Ehrig</a>; Gregor Engels; <a href="/facts/Hans-J%C3%B6rg_Kreowski/JRSm2nGy">Hans-Jörg Kreowski</a>; Grzegorz Rozenberg, eds. (Oct 1999). <a href="https://www.worldscientific.com/worldscibooks/10.1142/4180"><i>Applications, Languages and Tools</i></a>. Handbook of Graph Grammars and Computing by Graph Transformation. Vol. 2. World Scientific. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F4180">10.1142/4180</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-4020-2.</li> <li>Hartmut Ehrig; Hans-Jörg Kreowski; Ugo Montanari; Grzegorz Rozenberg, eds. (Aug 1999). <a href="https://www.worldscientific.com/worldscibooks/10.1142/4181"><i>Concurrency, Parallelism, and Distribution</i></a>. Handbook of Graph Grammars and Computing by Graph Transformation. Vol. 3. World Scientific. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F4181">10.1142/4181</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-4021-9.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Perez 2009 covers this approach in detail. - Perez, P.P. (2009), Matrix Graph Grammars: An Algebraic Approach to Graph Dynamics, VDM Verlag, ISBN 978-3-639-21255-6 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"A Graph-Oriented Object Model for Database End-User Interfaces" (PDF). <a href="https://www.cs.indiana.edu/~vgucht/p24-gyssens.pdf" target="_blank">https://www.cs.indiana.edu/~vgucht/p24-gyssens.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"TERMGRAPH". <a href="http://www.termgraph.org.uk/" target="_blank">http://www.termgraph.org.uk/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>