Suspension (topology)

Graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering (software construction and also software verification) to layout algorithms and picture generation.

Graph transformations can be used as a computation abstraction. The basic idea is that the state of a computation can be represented as a graph, further steps in that computation can then be represented as transformation rules on that graph. Such rules consist of an original graph, which is to be matched to a subgraph in the complete state, and a replacing graph, which will replace the matched subgraph.

Formally, a graph rewriting system usually consists of a set of graph rewrite rules of the form $L\rightarrow R$, 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 (pattern matching, thus solving the subgraph 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 languages; the different wording is used to emphasize the goal of constructions like the enumeration all graphs from some starting graph, i.e. the generation of a graph language – instead of simply 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. The algebraic approach is divided into some sub approaches, the double-pushout (DPO) approach and the single-pushout (SPO) approach being the most common ones; further on there are the sesqui-pushout and the pullback approach.

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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: $r=(L\leftarrow K\rightarrow R)$ (or $L\supseteq K\subseteq R$) where $K\rightarrow L$ 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 $k\colon K\rightarrow G$ (this is where the name double-pushout comes from). Another graph morphism $m\colon L\rightarrow G$ 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 $G$ (see subgraph isomorphism problem), and after a match is found, $L$ is replaced with $R$ in host graph $G$ where $K$ serves as an interface, containing the nodes and edges which are preserved when applying the rule. The graph $K$ 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 $G$.

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: $r\colon L\rightarrow R$. 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 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.

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 focuses entirely on research into term graph rewriting and its applications.

Implementations and applications

Example for graph rewrite rule (optimization from compiler construction: multiplication with 2 replaced by addition)

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.

• Tools that are application domain neutral:
  • GrGen.NET, the graph rewrite generator, a graph transformation tool emitting C#-code or .NET-assemblies
  • AGG, the attributed graph grammar system (Java)
  • GP (Graph Programs) is a programming language for computing on graphs by the directed application of graph transformation rules.
  • GMTE, the Graph Matching and Transformation Engine for graph matching and transformation. It is an implementation of an extension of Messmer’s algorithm using C++.
  • GROOVE, 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.
• Tools that solve software engineering tasks (mainly MDA) with graph rewriting:
  • GReAT
  • VIATRA
  • Graph databases often support dynamic rewriting of graphs
  • Gremlin, a graph-based programming language (see Graph Rewriting)
  • PROGRES, an integrated environment and very high level language for PROgrammed Graph REwriting Systems
  • Fujaba uses Story driven modelling, a graph rewrite language based on PROGRES
  • EMorF and Henshin, graph rewriting systems based on EMF, supporting in-place model transformation and model to model transformation
• Mechanical engineering
  • GraphSynth 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 XML files and is written in C#.
  • booggie integrates GrGen.NET with a port-based metamodel definition and an OpenGL graph visualization based on Tulip
• Biology
  • Design and implementation of a graph grammar based language for functional-structural plant modeling (other domains: graphics, architecture)
• Artificial Intelligence/Natural Language Processing
  • OpenCog provides a basic pattern matcher (on hypergraphs) which is used to implement various AI algorithms.
  • RelEx is an English-language parser that employs graph re-writing to convert a link parse into a dependency parse.

See also

• Category theory
• Graph theory
• Shape grammar
• Term graph

Notes

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References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Heckel, R. (2006). Graph transformation in a nutshell. Electronic Notes in Theoretical Computer Science 148 (1 SPEC. ISS.), pp. 187–198.
• König, Barbara (December 2004). Analysis and Verification of Systems with Dynamically Evolving Structure. Habilitation thesis, Universität Stuttgart, pp. 65–180.