Survey of Graph Rewriting applied to Model Transformations

Francisco de la Parra and Thomas Dean

School of Computing, Queen’s University, Kingston, Ontario, Canada

Keywords:

Graph Grammars, Graph Rewriting, Model Transformations, Domain Analysis, Model-Driven Engineering.

Abstract:

Model-based software development has become a mainstream approach for efﬁciently producing designs, test

suites and program code. In this context, model-to-model transformations have become ﬁrst-class entities and

their classiﬁcation, formalization and implementation are the subject of ongoing research. This work surveys

the characteristics and properties of graph rewriting systems and their application to formalize and implement

transformations of language-based software models. A model’s structure or behaviour can be abstracted into

the deﬁnition of a given graph type. Its structural and behavioural changes can be represented by rule-based

transformation of graphs.

1 INTRODUCTION

Model Driven Engineering (MDE) has become a

sound and efﬁcient alternative for the design and de-

velopment of complex software of the kind found in

industrial and networked applications. Efﬁcient ap-

plication of MDE can produce powerful and reusable

designs, as well as readily veriﬁable program code

(Neema and Karsai, 2006).

In MDE, model-to-model transformations are

ﬁrst-class entities and their classiﬁcation, formaliza-

tion and research has focused on: 1) analyzing their

intrinsic properties; 2) deﬁning formal syntactic and

semantic aspects of their realization; and 3) devel-

oping approaches for their implementation (B

ezivin,

2004; Czarnecki and Helsen, 2006; Mens and Gorp,

2006).

Many modeling languages utilize graphs in the

core of their representation. A model’s structure can

be abstracted as a given graph type. Its structural

and behavioural changes can be represented by rule-

based transformations of graphs (Boyd and McBrien,

2005; Karsai, 2010). This work surveys the char-

acteristics of graph rewriting systems, known as: 1)

graph grammars if the emphasis is on studying the

types of graphs they generate (i.e., graph languages),

2) graph transformation systems if the primary inter-

est is in describing the graph rewriting mechanisms

they use; and their application to formalize, specify

and implement transformations on models. The pre-

sentation focuses on providing a concise and system-

oriented view of the more versatile graph rewriting

techniques which could aid MDE researchers in de-

vising rule-based model transformations.

1.1 Background

Graph grammars and graph transformations have

been used in software engineering as formalisms for

representing design and computational aspects of sys-

tems at a very high level of abstraction.

Application Domains and Modeling. Application

domain analysis (Prieto-D

ıaz, 1990) provides precise

descriptions and formalizations of the objects and ac-

tivities in a given structured environment. Graphs and

graph transformations are a capable meta-language

to abstract from real systems or other models (Kent,

2002; K

uhne, 2006). In the language-oriented con-

text of MDE, they represent a promisory alternative

to provide more precise syntactic and semantic deﬁ-

nitions of diagrammatic modeling languages.

Graph Grammars. From its intial introduction in

the late 1960’s, graph grammar theory has developed

into a number of approaches: algebraic techniques,

set theoretic approaches, node-label controlled (NLC)

approaches, etc. (Ehrig et al., 1991; Nagl, 1987;

Rozenberg, 1997; Sch

urr, 1995). From a software en-

gineering perspective, the initial focus on graph gram-

mars has shifted from understanding the properties

of the graph languages they generate to an interest

in describing and exploiting the graph transformation

mechanisms they use, how they integrate into graph

431

de la Parra F. and Dean T..

Survey of Graph Rewriting applied to Model Transformations.

DOI: 10.5220/0004731504310441

In Proceedings of the 2nd International Conference on Model-Driven Engineering and Software Development (MODELSWARD-2014), pages 431-441

ISBN: 978-989-758-007-9

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

rewriting systems, and how they can be applied as

high level speciﬁcations in modeling systems and pro-

cesses.

Graph Rewriting Tools. Graph

(Himsolt, 1991)

and Algebraic Graph Grammar (AGG) system

(Taentzer, 2004) represent early efforts to prototype

graph grammars in a standalone fashion. PROGRES

was one of the ﬁrst integrated environments for in-

teractively prototyping graph grammars and transfor-

mations with the goal of producing software artifacts

(Nagl and Sch

urr, 1991; Sch

urr et al., 1995). The en-

vironment consisted of: 1) a visual/operational speci-

ﬁcation language for graphs and graph rewriting rules,

2) a set of tools for editing, analyzing and execut-

ing speciﬁcations, and 3) a methodology to use of

the language and tools, denominated graph grammar

engineering. PROGRES can be considered a pre-

cursor for a number of more specialized prototyp-

ing tools dealing with models and model transfor-

mations: AToM

(de Lara et al., 2004b), the Graph

Rewriting and Transformation (GReAT) system (Bal-

asubramanian et al., 2006; Karsai et al., 2003), FU-

JABA (Nickel et al., 2000), MOFLON (Weisem

oller

et al., 2011), VIATRA2 (Balogh and Varr

o, 2006),

and MATE (Legros et al., 2011).

2 GRAPH GRAMMAR

APPROACHES

2.1 General Concepts

The following is an abbreviated description of key

components.

2.1.1 Graphs and Hypergraphs

A simple labeled directed graph G over a pair of

label alphabets (Σ

, Σ

) (where Σ

= {vl

, ..., vl

node label alphabet, and Σ

= {el

, ..., el

}: edge la-

bel alphabet) consists of a sextuple (V, E, s

, t

, l

)

where V = {v

, ..., v

} (set of nodes; or vertices),

E = {e

, ..., e

} (set of directed edges), s

: E → V

assigns an edge’s source node, t

: E → V assigns an

edge’s target node, l

: V → Σ

assigns node labels,

: E → Σ

assigns edge labels (ﬁg. 1-(a)).

From the previous deﬁnition of a simple labeled

graph, one can derive the more speciﬁc classes of

graphs: 1) undirected graphs, if for all edges the

source and target are not identiﬁed; 2) multigraphs

multiple edges can be deﬁned between a pair of

source and target nodes; 3) unlabeled graph, if the la-

bel alphabets Σ

, Σ

are empty sets; 4) typed graph,

Figure 1: Graphs and Hypergraphs.

if the sets of node and edge labels are partitioned into

classes called types; 5) attributed graph, if nodes are

assigned attributes of a given data type; and 6) Star

graphs consisting of nonterminal and terminal nodes,

where the node labeled as nonterminal is at the center

of a star conﬁguration (ﬁg. 1-(b)).

Hypergraphs represent an extension of the struc-

tural concept of a graph to contain hyperedges, in-

stead of edges, connecting to multiple source and tar-

get nodes (ﬁg. 1-(c)) (Habel, 1992).

Triples of Graphs extend the idea of a single graph

as the unit of analysis to a triple (LG, CG, RG) con-

sisting of three graphs: LG (left graph), RG (right

graph), and CG (connection graph), of the same class,

where graph CG represents associations between the

elements of graphs LG and RG. Triple graph gram-

mars are the structures associated with generating and

parsing these graph triples (Sch

urr, 1995).

Hierarchical Graphs constitute a most general

structure with higher-order nodes that are abstrac-

tions of graphs and higher-order edges that repre-

sent relations between graphs which are represented

as graphs again. With these structures, it is possible

to abstract an entire sub-graph to a node or to a single

edge between two abstracted nodes (Dean and Cordy,

1995).

2.1.2 Graph Rewriting Productions

The following is a general deﬁnition of a production

that is applicable to most graph transformation (i.e.,

graph grammar) approaches found in the literature

(Andries et al., 1999).

A graph rewriting production is a sextuple p =

(LG, K, RG, ac, gl, em)

that allows the transforma-

tion of a host graph G into a direct derivation graph

H, whose components are deﬁned as follows:

LG is the production’s left graph. Applying a pro-

duction to a host graph G involves searching for one

or more isomorphic occurrences of LG in G which

will eventually be replaced by instantiations of graph

RG.

A production either uses embedding or gluing but not

both simultaneously

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K is the production’s interface graph which is a

sub-graph (i.e., no dangling edges: all its edges have

a target and a source node). LG and RG contain an

isomorphic occurrence of it.

RG is the production’s right graph which will be

attached to an intermediate graph (G − LG) ∪ K, in

the more common case, using graph gluing or embed-

ding, or both operations, to produce a direct derivation

graph H as the overall result of applying the produc-

tion.

ac are the application conditions under which a

production will be applied. They could represent the

existence or non-existence of certain nodes, edges, or

subgraphs in the host graph G, as well as embedding

restrictions of LG in G or of RG in H.

gl is the gluing operation which consists of three

steps: 1) Build a context graph D which contains the

graph G, minus the nodes and edges of the isomorphic

occurrence of LG in G that are not preserved in RG; 2)

Glue graphs D and LG through K to generate the host

graph G. This operation is the disjoint union of graphs

D and LG; and 3) Glue graphs D and RG through K to

generate the direct derivation graph H. This operation

is the disjoint union of graphs D and RG.

em is the embedding (or connecting) operation,

which creates edges between designated nodes in the

context graph D and nodes in the graph RG, to pro-

duce the ﬁnal direct derivation graph H.

2.1.3 Graph Grammars

A graph grammar is a quadruple GG = (T, S, P, A)

where:

T is the type or class of graph (e.g., labeled, hy-

pergraph), that can be generated, modiﬁed and recog-

nized by applying a set P of productions.

S is the start graph to which a set P of productions

will be initially applied.

P is a set of productions P = {p

, ..., p

A is a set of additional speciﬁcations extending the

scope and nature of applying the set of productions

P to graphs of type T (e.g., attributes, programmed

rules, global application conditions, structural condi-

tions).

The language L(GG) generated by graph grammar

GG is the set of terminal graphs that can be generated

from the start graph S by repeated application of the

graph rewriting productions.

Applying a set of productions P = {p

, ..., p

}

to one or more graphs involves scheduling the ap-

plication of the individual productions in a non-

deterministic or deterministic manner. Determinis-

tic approaches can utilize ad-hoc programmed rules,

prioritization schemes, application conditions and hy-

brid schemes to schedule productions. It is well

known that these extensions reduce the generative

power of a grammar, making the parsing and recog-

nition of the graphs in the generated language a more

manageable task (Nagl, 1987). In general, they fa-

cilitate using graph grammars in applications such as

subject modeling, entity description, process speciﬁ-

cation and pattern recognition, by increasing their de-

scriptive power (Drewes et al., 2008).

2.2 Context-free Graph Grammars

Context-free grammars replace a single node or edge

by a complete graph when applying a production.

They are important in that graph derivations can be

modelled by derivation trees which produce the same

set of graphs for any different ordering used to apply

the productions (Courcelle, 1987). They are useful in

describing and generating graphs with recursive prop-

erties.

2.2.1 Algorithmic Approaches

These approaches to graph grammars implement a

graph rewriting mechanism which replaces a node v

labeled with the symbol l from an alphabet Σ, in a

host graph G, by reconnecting to the context graph

D = G − v

− E

: set of dangling edges in G af-

ter removing v

) an isomorphic copy of a grammar

production’s RG graph through edges determined by

path expressions (Nagl, 1987), connection relations

(Rozenberg, 1997), or some other algorithm.

The following three sections introduce the gram-

mar types: NLC, NCE and edNCE, which have com-

pletely local graph rewriting mechanisms, where a

production’s RG graph is reconnected only to the

neighbour nodes of the node being replaced.

NLC Grammars. A node-label-controlled (NLC)

graph grammar is a quadruple (NU, P, C, S) where:

NU = (V, E, Σ, L

) is a node-labeled undirected

graph type where: 1) V is a ﬁnite set of nodes; 2)

E is a set of undirected edges; 3) Σ = Σ

∪ Σ

is a ﬁ-

nite alphabet of symbols, with Σ

: non-terminals, Σ

terminals; and 4) l

: V → Σ is the labelling function

which assigns a terminal or non-terminal symbol to

each node in V .

P = {p

, ..., p

} is a ﬁnite set of productions, with

each p

= (L

, RG

, C

), L

: non-terminal symbol in Σ

(left side of production), RG

: graph of type NU (right

side of production), C

: connection relation of p

C = {(ρ, σ)|ρ, σ ∈ Σ} =

1≤i≤n

(C ⊆ Σ × Σ) is

the connection relation. A pair (ρ, σ) in this relation

indicates that an edge should be created between each

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433

node labeled σ in graph RG and each node labeled ρ

in the context graph D.

S is the start graph, usually a single node labeled

with a non-terminal symbol.

The graph replacement and embedding mecha-

nisms in NLC grammars are strictly local to the vicin-

ity of a single labeled node.

NCE Grammars. A neighbourhood controlled em-

bedding (NCE) grammar represents a reﬁnement of a

NLC grammar to provide enhanced control over the

local graph embedding mechanism. It is deﬁned as

a quadruple (NU, P, C, S) where: 1) NU as in a NLC

grammar; 2) P is a set of productions as speciﬁed for

NLC grammars, with the addition that the right-hand-

sides of the pairs in the connection relation C

of each

production now refer to speciﬁc nodes in graph RG;

3) C = {(ρ, v)|ρ ∈ Σ, v ∈ V } =

1≤i≤n

(C ⊆ Σ×V )

is the connection relation. A pair (ρ, v) in this rela-

tion indicates that an edge should be created between

a speciﬁc node labeled v in one of the productions

and any node labeled ρ in the context graph; and

4) S is the start graph as deﬁned in a NLC gram-

mar. These grammars still generate undirected node-

labeled graphs with unlabeled edges.

edNCE Grammars. Edge-labeled directed-graph

neighbourhood controlled embedding (edNCE) gram-

mars represent an improvement of NLC grammars

with an enhanced graph embedding mechanism and

enhanced structural rules to generate and recognize

more general classes of graphs. An edNCE gram-

mar is a quadruple (ND, P, C, S) where: 1) ND is de-

ﬁned similarly to NU with the enhancement of la-

beled directed edges; 2) P is a ﬁnite set of produc-

tions whose left-hand-sides are deﬁned as in the case

of NLC grammars, whose right-hand-sides are now

graphs of type ND, and the right-hand-sides of the

pairs in the connection relation C

refer now to spe-

ciﬁc nodes in graph RG; 3) C = {(ρ, v, d)|ρ ∈ Σ, v ∈

V, d ∈ {in, out}} =

1≤i≤n

(C ⊆ Σ ×V ) is the con-

nection relation, a triple (ρ, v, d) in this relation indi-

cates that an incoming (d = in) or outgoing (d = out)

directed edge should be created between a speciﬁc

node labeled v in one of the productions and any node

labeled ρ in the context graph; and 4) S is the start

graph as deﬁned for a NLC grammar.

It is well known that some other possible exten-

sions to the graph embedding mechanism, for exam-

ple, allowing ﬂipping directions on edges, or allowing

multiple edges in the tuples of the connection relation,

do not increase the generation power of an edNCE

grammar (Rozenberg, 1997).

Grammars with Arbitrary Embedding. In gen-

eral, the embedding mechanism of a production’s RG

graph can be extended to an arbitrary degree of com-

plexity, beyond local schemes, allowing reconnecting

it to not only neighbour nodes of the node being re-

placed but also to any node in the context graph D.

This would allow a further increase in the generation

power of a grammar using the node reconnecting ap-

proach. However, the trade-off is increased complex-

ity of the generated graphs, which could make parsing

them a very difﬁcult or even intractable task. Further-

more, applying these grammars to modeling could be-

come cumbersome.

The algorithmic approach to grammars can be

used to describe the syntax of simple diagrams such

as object models in UML.

2.2.2 Hypergraph Grammars

The central element of a hypergraph is the hyper-

edge, which is an abstraction consisting of a sin-

gle edge h with N links (“tentacles”) that are con-

nected to a set S(h) of m source nodes and a set

T (h) of n target nodes. Similar to ordinary graphs,

one can deﬁne different types of hypergraphs: un-

labeled, directed labeled, etc. A special type is a

directed labeled multi-pointed hypergraph, denomi-

nated an (m, n)-hypergraph, which contains two sets

of labeled sequential nodes: begin and end, with car-

dinalities m and n, representing overall source and tar-

get nodes of the hypergraph, respectively. This struc-

ture provides the elements to implement hyperedge

replacement grammar productions using a node glu-

ing approach, where a single hyperedge is replaced by

a (m,n)-hypergraph. The gluing operation occurs be-

tween the hyperedge’s S(h) and T (h) nodes and sets

begin and end of a (m,n)-hypergraph, respectively.

Hyperedge-Replacement Grammars. A hyper-

edge replacement (HR) grammar is a quadruple

, P, C, S) where: 1) H

is a (m,n)-hypergraph

speciﬁcation; 2) P is a set of productions p

, each

= (L

, RH

, C

), L

is the label of a single hyperedge

h (production’s left side), RH

: hypergraph of type H

(production’s right side), C

: re-connection mapping

for RH

, production p

applied to a host hypergraph

GH replaces hyperedge L

by hypergraph RH

recon-

necting it according to mapping C

to produce derived

hypergraph HH; 3) C = {(δ

, δ

) | δ

: π(begin) →

S(h), δ

: π(end) → T (h); δ

, δ

surjective functions}

(π(begin), π(end): permutations of begin and end, re-

spectively. C =

1≤i≤n

) is the node-gluing map-

ping; 4) S is the start hypergraph, usually a single

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handle

labeled with a non-terminal symbol.

HR-grammars have convenient context-free prop-

erties, mostly due to the fact that their hypergraph re-

placement mechanism is completely local to a single

hyperedge. They are quite ﬂexible structures with a

wide spectrum of generation power, considering that

they can generate graphs and hypergraphs of varying

complexity just by changing the type (i.e., number of

tentacles) on hyperedges. The hyperedge concept is a

fundamental structure in many different types of mod-

els (Habel, 1992), hence it is applicable to consider

HR-grammars as having a wide range of descriptive

power. For example, transitions in Petri nets are more

easily represented as hypergraphs than as complex

subgraphs.

Hypergraph Replacement Grammars. In these

structures, denominated HGR-grammars, productions

can replace arbitrary sub-hypergraphs in a given host

hypergraph. They use the double pushout graph

rewriting technique deﬁned in the algebraic approach

to graph grammars (Section 2.3) to generate derived

hypergraphs. HGR-grammars can provide more ﬂexi-

ble and less granular productions than HR-grammars,

and can be reduced to the latter grammars when the

sub-hypergraph replaced is a single hyperedge.

The generative power of HR- and HGR-grammars

and the classes of languages they generate have been

extensively studied (Habel, 1992; Rozenberg, 1997),

however, the attention has shifted to studying speciﬁc

hypergraphs such as jungles

(Ehrig et al., 1999) and

their rewriting mechanisms. The nodes and edges of a

jungle can represent the terms and operations of func-

tional or algebraic expressions. This approach allows

a more efﬁcient evaluation of expressions by avoiding

the multiple representation, hence the re-calculation,

of repeated terms.

2.3 Algebraic Approach to Graph

Grammars

The algebraic approach to graph grammars focuses

on the rewriting mechanism of direct derivations of

graphs, rather than on the study of general constructs

to obtain graph languages (Ehrig, 1979; Ehrig et al.,

2006; Rozenberg, 1997). Its basic operation is graph

gluing, where common nodes and edges in a produc-

tion’s LG (left) and RG (right) graphs play a key role

in maintaining graph consistency when replacing LG

A handle is a single (m, n)-hyperedge connected to sets

of nodes begin and end.

jungles are forests of coalesced trees with shared struc-

tures that can be represented by acyclic hypergraphs

Figure 2: Single and Double Pushout Production Applica-

tion (adapted from (Ehrig et al., 2006)).

by RG in a host graph G. In this approach, given a

node- and edge-labeled simple graph G, the applica-

tion of a production p yields a graph H if the follow-

ing conditions are met:

1) The deﬁnition of production p is based on par-

tial graph morphisms between its LG (left graph) and

its RG (right graph) in the case of the single pushout

approach (SPO) (ﬁg. 2-(b)), or is based on total graph

morphisms between LG, RG and an auxiliary inter-

face graph K, containing the nodes and edges required

for gluing RG to a context graph D (ﬁg. 2-(a)), in the

case of the double pushout approach (DPO).

2) The application of production p, represented by

pushout constructions (1) and (2), or (3) in the DPO

and SPO approaches, respectively, removes the nodes

and edges of G that can be mapped to LG but not to

RG, and adds to G the new nodes and edges of RG to

produce H.

Similar to all approaches, a graph grammar GG

consists of a start graph G

and a ﬁnite set of pro-

ductions P = {p

, ... p

}, each production as deﬁned

above. The language L(GG) generated by the gram-

mar is the set {G

| G

∗

⇒ G

;n = 1, 2, ...}.

The grammars obtained using this approach,

generally context-sensitive, can be highly tuned to

speciﬁc graph modeling situations with arbitrary

start graphs and application-speciﬁc graph derivation

steps.

The focus on direct graph derivations, typical of

the algebraic approach, facilitates detailed analyses

of comparative properties, structural transformations,

and semantics associated with speciﬁc subsets of pro-

ductions in a grammar and on speciﬁc graphs of its

generated language, which can be used to ﬁnd opti-

mized ways to transform graphs of a given class.

If two productions p

and p

must work coopera-

tively on graphs to model, for example, a concept of

synchronized execution of tasks or updates to systems

states, then they must synchronize their application to

common items accessed by both on those graphs. One

solution to this problem is to amalgamate the shared

SurveyofGraphRewritingappliedtoModelTransformations

435

effects of both productions into a common subproduc-

tion p

, embedded in p

and p

If a derivation frequently occurs in some model-

ing situation and the interest concentrates on the ini-

tial and ﬁnal graphs, it seems natural to search for

an optimization of the productions that would by-

pass the derivation of the intermediate graphs. This

problem translates to ﬁnding a derived production

∗

: G

. This production exists if, given a

derivation α ≡ G

⇒ ...

⇒ G

and an injective mor-

phism G

→ H

, e

induces an embedding e : α → β

(where β ≡ H

⇒ ...

⇒ H

) and H

∗

⇒ H

. De-

rived production p

∗

for derivation sequence α is given

by p

∗

= p

∗

;...; p

∗

(i.e., sequential application of co-

productions p

∗

(ﬁg. 2-(b))).

2.4 Triple Graph Grammars

The formalism of triple graph grammars (TGG) was

devised (Sch

urr, 1995) to generate and transform

pairs of related graphs (usually called source and

target graphs) under a well-formed synchronization

mechanism (i.e., a connection graph) which would

preserve the relations in the pair after applying a

transformation. The initial motivation of the formal-

ism was to provide a high-level graph-based model-

ing and speciﬁcation tool for problems involving re-

lated diagrams (i.e., syntax trees, control ﬂow dia-

grams) and information structures (i.e., requirements,

design and traceability documents) requiring simul-

taneous update for consistency purposes. More re-

cently, key concepts found in TGGs have been used

in the OMG’s QVT model transformation language

standard (Object Management Group, 2011) to de-

ﬁne the ”mappings” of a transformation. Also, appli-

cations of TGGs to the speciﬁcation of bidirectional

and incremental model transformations are becoming

more common (Czarnecki et al., 2009; Stevens, 2010;

Hermann et al., 2013).

The categorical framework (Ehrig et al., 2006)

is the standard mathematical tool used to describe

TGGs: their elements and operations. TGG’s ele-

mentary unit of transformation is the graph triple, de-

noted as GT ≡ (LG

← CG

→ RG), where LG: left

graph, CG: connection graph, RG: right graph (usu-

ally node- and edge-labeled directed graphs, however,

other types of graphs can be used), and gl, gr are

graph morphisms (ﬁg. 3). The transformation oper-

ator is the triple production, denoted as p ≡ (l p

←

→ rp), and consisting of three monotonic sin-

gle productions

: l p ≡ (LL, LR), cp ≡ (CL, CR) and

Production p ≡ (L, R) is monotonic if L ⊆ R.

Figure 3: TGG Production Application (adapted from

(Sch

urr, 1995)).

rp ≡ (RL, RR) (with graph morphisms l p : LL → LR,

cp : CL → CR and rp : RL → RR, respectively, and

the induced graph morphisms lr|

: CL → LL, rr|

CL → RL

, lr : CR → LR and rr : CR → RR), which

are applied simultaneously (ﬁg. 3).

Applying a triple production p to a graph triple

GT produces a directly derived triple HT ≡ (LH

←

→ RH) (ﬁg. 3), derivation denoted as GT

HT ,

where the application of each of its component pro-

ductions l p, cp and rp are modeled by the single

pushout categorical construct described in ﬁg. 2-(b).

That is, productions l p, cp and r p, match LL, CL and

RL in graphs LG, CG and RG through total morphisms

lg, cg and rg, respectively. Then, they transform the

latter triple into graphs LH, CH and RH, which are

related to the applied production and original graphs

by the morphism pairs (lh, l p*), (ch, cp*), and (rh,

rp*), respectively (ﬁg. 3).

The basic deﬁnition of TGGs with monotonic pro-

ductions, although partially restrictive, makes it more

feasible to specify translation tools based on inter-

graph relationships by simplifying the manipulations

required to obtain a triple’s target graph that an arbi-

trary source graph, as shown below (Sch

urr, 1995).

A triple production p ≡ ((LL, LR)

← (CL, CR)

→

(RL, RR)) can be split into a pair of equivalent pro-

ductions: a left-local production p

and a left-to-right

production p

, with:

• p

≡ ((LL, LR)

← (φ, φ)

→ (φ, φ))

• p

≡ ((LR, LR)

← (CL, CR)

→ (RL, RR))

When p is applied to a triple GT , of which, only

graph LG is known, denoted by GT

p(lh)

HT (HT : de-

rived triple), if its component productions are mono-

tonic, the following equivalence holds:

lr|

and rr|

indicate that lr and rr are total mor-

phisms over graph CL

φ: empty graph; ε: inclusion of empty graph in any

graph.

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p(lh)

HT ⇐⇒ ∃ XT | GT

(lh)

XT ∧

(lh)

This result extends to the application of a se-

quence of triple productions p

, ..., p

in the form of

the following equation:

(lh

) ◦ ... ◦ p

(lh

) =

(lh

)◦... ◦ p

))◦(p

(lh

)◦... ◦ p

(lh

))

which translates to say that the application of a se-

quence of triple productions is equivalent to the ap-

plication of a sequence of left-local productions fol-

lowed by the application of a sequence of left-to-right

productions. The previous analysis is also valid for

right-local and right-to-left productions.

3 GRAPH TRANSFORMATION

SYSTEMS

A signiﬁcant body of concepts is common to all the

graph grammar approaches found in the literature.

Graph types and production types are common ex-

amples of useful abstractions. When extending these

ideas to applications involving large numbers of rules

(i.e., productions) and complex systems, resorting to

proven software engineering principles (i.e., classi-

ﬁcation, encapsulation, reuse and module-based se-

mantics) to further structure the graph transformation

process is essential if the requirement is to produce

manageable and scalable graph-based systems.

Graph transformation systems (Ehrig et al., 1999;

Kreowski et al., 2010) is a formalism that focus on

deﬁning encapsulation abstractions such as transfor-

mation units and modules, using precise graph-based

semantics, which can be reused and aggregated to

build larger systems. The basic structuring gener-

alization is a graph transformation approach A =

(G, R , ⇒, E , C ), where:

G is a class of graphs of a speciﬁed type (i.e., la-

belled graphs, hypergraphs, etc.).

R is a class of rules with the same structure (i.e.,

single pushout, double pushout, etc.).

⇒ is a rule application operator that for each rule

r ∈ R produces a relation

⇒≡ {(g

, g

) ∈ G × G | g

⇒ g

E is a class of graph class expressions, where each

one of them deﬁnes: a ﬁnite enumeration of graphs,

or a set theoretic deﬁnition of graphs, or a graph the-

oretic property, or the exhaustive output of a given

transformation unit, or some graph meta-deﬁnition

scheme, or the result of boolean operations on graph

classes.

C is a class of elementary control conditions over

a set ID of identiﬁers, which are expressed as rule ex-

ecution schedules, priority schemes, boolean expres-

sions and regular languages. For a given environment

E, deﬁned by the mapping E : ID → 2

G×G 7

, each

c ∈ C speciﬁes a relation SEM

(G, G

) ∈ G × G, where G: initial graph, G

: derived

graph.

This scheme allows to refer to the multiple gram-

mar types in an approach-independent manner that fa-

cilitates the analysis of similarities and differences in

properties, structures and rules.

Given a graph transformation approach, a trans-

formation system consists of a set of transformation

units with the capability to import each other forming

acyclic or cyclic import paths. Each transformation

unit produces an output terminal graph from an ini-

tial graph by interleaving rule applications with calls

to other transformation units, where the latter are ex-

ecuted in an atomic manner.

3.1 Transformation Units

A transformation unit (TU) consists of a set of graph

transformation rules conforming to a graph transfor-

mation approach A and a set of atomic-execution calls

to imported transformation units, where elements of

both sets execute interleaving each other (”interleav-

ing semantics”) to transform an initial graph of a

given class into a speciﬁc terminal graph of the same

or another class. The calls to other import transfor-

mation units can be cyclic or acyclic, depending on

whether a “called unit” calls one of its “calling units”

in a given calling path or not.

The input and output

graph classes are deﬁned by class expressions.

A TU can also include a control condition (cc)

whose purpose is to regulate, restrict, and possibly

eliminate the inherent non-determinism appearing in

graph transformations due to the fact that: 1) multi-

ple rules can simultaneously be applicable to an ex-

isting graph, 2) there can exist multiple places in a

graph where a given rule could be applied. A cc

can have properties of minimality (i.e., the graphs

transformed by a TU are only the ones allowed by

the cc), invertibility (i.e., for a given cc C, C

−1

ex-

ists) and continuity (i.e., given a TU with a cc C

over a sequence of environments {E

, ..., E

}, then

SEM

∪...∪E

(C)∪... ∪SEM

(C)). The

following are the most common types of control con-

ditions:

1) Control conditions of language type which only

allow interleaving sequences of rules and imported

Given a set A, 2

denotes its power set

This analysis assumes acyclic calling paths.

SurveyofGraphRewritingappliedtoModelTransformations

437

TUs that can be represented by strings belonging to

a speciﬁc control language L.

2) Priorities assigned to the rules of a TU can rep-

resent control conditions that allow only speciﬁc pairs

(G, G

) ∈ SEM

(C).

3) Reduced graphs with respect to a control condi-

tion (denoted C!) is a type of control condition where

the only pairs (G, G

) ∈ G ×G allowed, have the prop-

erty that there is no graph G

such that (G

, G

) ∈

SEM

(C!).

4) Rules applied as-long-as-possible (denoted R!)

for a given rule set R constitutes a special type of re-

duced graphs control condition.

5) A TU can be a control condition as it deﬁnes a

binary relation on graphs.

As a generative structure, a TU over a graph

transformation approach A is a 5-tuple TU =

(IG, L, R, CC, T G), where: 1) IG and T G are the

initial and terminal graph class expressions, respec-

tively; 2) L is a ﬁnite set of identiﬁers to refer to im-

ported transformation units; 3) R is a ﬁnite set of la-

belled rules (R ⊆ R ), whose identiﬁers are not in L;

and 4) CC is a control condition.

A TU has a graph-based interleaving operational

semantics, denoted as SEM(TU), implemented by:

1) a function SEM : L → 2

G×G

capable of assign-

ing a set of graphs to each identiﬁer associated with

a TU, either a graph transformation rule or an im-

ported transformation unit, 2) a set of graphs resulting

from applying the rules from R. The overall semantics

SEM(TU) of a TU is also a binary relation on graphs

having pairs (G, G

) where G and G

are compliant

with the initial and terminal graph class expressions,

respectively, and the control condition CC.

3.2 Transformation Modules

A transformation module (Ehrig et al., 1999) is use-

ful in the speciﬁcation of very large systems to en-

capsulate a set of transformation units, making some

hidden from the outside and some visible through

an export interface. Formally, it is deﬁned over a

graph transformation approach A

as a triple MOD =

(IMPORT, BODY, EX PORT ) where: 1) IMPORT is

a set of imported names referring to external TUs; 2)

BODY and is a ﬁnite set of local named TUs, each

of which can use TUs from sets BODY and IMPORT

only, and named rules over some set L of names; and

3) EXPORT is a set of exported names referring to

TUs in BODY and/or IMPORT . The following re-

strictions apply: 1) Imported names cannot be reused

for a local rule or TU (i.e., L ∩ IMPORT = φ); and 2)

The set of all transformation modules over a graph

transformation approach A is denoted τ

Figure 4: Modeling and Model Transformation Architec-

ture.

Local TUs use only names that are imported or repre-

sent local TUs to refer to imported TUs.

The semantics of a module over a graph transfor-

mation approach A, denoted as

SEM

mod

(MOD), deﬁnes an environment E

mod

that

associates binary relations on graphs to each imported

name, each local rule, each locally deﬁned TU, and

each exported name in the EXPORT set. The export

semantics of a module is obtained as the restriction of

mod

to names in EX PORT only.

4 GRAPH-BASED MODEL

TRANSFORMATION

The suitable use of graphs and graph transformations

for representing models and model transformations

could potentially provide the tools to deﬁne domain

modeling and transformations languages with very

precise syntax and semantics.

4.1 Model Transformations

Reliable transformations on language-based models

have to correctly interpret the syntax and semantics of

the modeling language(s) deﬁning the source models,

and at the same time, correctly reproduce the syntax

and semantics of the modeling language(s). During

software development, the views of the system, rep-

resented by models, can change (i.e., model-to-model

transformations), and models are reﬁned to lower ab-

straction levels to generate system implementations.

The following model, adapted from (Metzger, 2005),

provides a unifying and systemic understanding of

these multiple options occurring in MDE develop-

ment, which would include graph languages: 1) A

modeling language (ML) is a formalism which pre-

cisely deﬁnes the notation and meaning of a model

(ﬁg. 4); 2) The ML’s notation consists of concrete

and abstract syntax; 3) The ML’s meaning consists of

the static and dynamic semantics.

Using the conceptual model of ﬁg. 4, one can de-

ﬁne a model transformation as a mapping:

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438

f : M

Source

(S)|

Source

→ M

Target

(S)|

Target

where ML

Source

and ML

Target

are the modeling lan-

guages deﬁning the source ML

Source

(S) and target

Target

(S) models of the system S, respectively.

Visualizing and interpreting some of the existing

relationships and elements in ﬁg.4, one can provide

a brief taxonomy of model transformations, based on

the more relevant properties, as follows:

1) Directionality: a unidirectional transforma-

tion only interprets its source and reproduces its

target models. A bidirectional transformation can

also switch interpretation to target and generation to

source models.

2) Degree of Evolution: a horizontal transforma-

tion basically generates an equivalent view of the sys-

tem at the same abstraction level, often using the same

source and target modeling languages. A vertical

transformation generally uses different modeling lan-

guages for source and target models, for example in

code generation, and reduces the abstraction level.

3) Implementation Type: a transformation can be

implemented using a declarative approach based on

rules and pre/post-conditions, or an operational ap-

proach which explicitly speciﬁes the sequence of ac-

tions to transform a model.

4) Atomicity: a transformation can be executed as

a single non-decomposable step or as multiple steps

allowing for partial rollbacks.

5) Degree of Automation: the conﬁguration and

execution of a transformation can be completely au-

tomated by a software tool, or it can be partially auto-

mated, requiring user intervention to complete some

manual steps.

Other classiﬁcations of transformations exist

(Czarnecki and Helsen, 2006; Metzger, 2005; Mens

and Gorp, 2006), but the properties considered tend

to be reﬁnements of the ﬁve above.

4.2 Model Transformations using

Graphs

Models in MDE routinely express complex structural

and dynamic aspects of a system by using fairly in-

tuitive visual languages. If one formalizes syntactic

and semantic aspects of these languages using graphs,

then one can make use of graph grammars and graph

transformations to specify precisely how these models

should be built and how they should be transformed

(Grunske et al., 2005).

Graph transformations, with their deﬁned graph

types and rule sets, provide a solid foundation for rea-

soning about structural, semantic and operational as-

pects of model transformations in the following roles:

• as a semantic domain that directly provides

a generic speciﬁcation language and semantic

model for very high level deﬁnitions of applica-

tion domains, their abstractions and the transfor-

mations of those abstractions to support software

development activities such as the speciﬁcation of

functional requirements and description of archi-

tectural changes (Buchmann et al., 2008; Grunske

et al., 2005).

• as a meta-language to be used in the formal spec-

iﬁcation of the syntax, semantic and manipulation

rules of the DSLs and model transformation lan-

guages (MTLs) deﬁning source/target models and

model transformations, respectively (Karsai et al.,

2003; de Lara et al., 2004a).

Graph-based Metamodeling. Metamodeling is a

widely accepted technique used to deﬁne the rules

of visual/diagrammatic languages, which provides a

ﬂexible conﬁguration environment, especially useful

when working with DSLs, and allows the checking of

whether a model produced in a speciﬁc language is

valid or not (Amelunxen et al., 2008; Levendovzky

et al., 2009). The construction of a metamodel re-

quires two basic elements: 1) a meta-language capa-

ble of describing the metamodel’s structure and rela-

tions between modeling items (e.g., class diagrams,

graphs) and 2) an instantiation relationship indicating

how model instances will be generated as a result of

using the metamodel.

In the context of graphs one can formally think

of metamodels as sets of typed graphs and of mod-

els as sets of instance graphs obtained from enacting

type-instance mappings from the former sets. Fur-

thermore, given a model of a certain type (e.g., dia-

grams, object structure, architectures), one can spec-

ify its transformations (e.g., visual/notational repre-

sentation changes, functional requirements, architec-

tural changes) as graph transformation rules over the

typed graphs of the associated metamodel. These

graph-based transformations are sufﬁciently generic

to: 1) specify syntactic changes when using different

source and target modeling languages and 2) deﬁne

semantic mappings when the target language is used

to deﬁne a semantic domain for the source language.

5 OPEN PROBLEMS

AND CONCLUSIONS

Research on the application of graph grammars and

graph transformations to model transformations is

currently fairly active, showing two main trends: 1)

SurveyofGraphRewritingappliedtoModelTransformations

439

mostly theoretical studies characterized by increas-

ingly abstract grammar-based formalisms employed

in analyzing and specifying high level properties of

model transformations, and 2) software engineering

perspectives interested in automating the implemen-

tation of model transformations occurring in speciﬁc

domains using graph-based speciﬁcations. There is

a noticeable gap between both trends that gives way

to an interesting set of open problems, among them:

1) Specialized graph structures that perform well in

the speciﬁcation and implementation of graph- and

metamodel-based model transformations would make

the declarative approach suitable for building com-

plex model transformations; 2) Most research proto-

typing tools are often cumbersome to use, offer solu-

tions for very narrow paradigms and incomplete sup-

port for systems integration and management. Mak-

ing graph transformations a solid tool in the context of

MDE requires further research into the requirements

that industrial strength graph-based tools should have,

given the foreseeable nature of applications in engi-

neering and business.

In general, this work has found that there is a large

gap between a theoretical body of numerous graph

grammar and graph transformation approaches and

its application to practical problems in software en-

gineering. Their use in formalizing syntactic and se-

mantic aspects of software artifacts still remains far

removed from proven software engineering method-

ologies and tools that have produced reliable, ef-

ﬁcient, user-friendly and maintainable applications.

The graph transformation research community has re-

cently become aware of these deﬁciencies and has

responded with studies placing the graph transfor-

mation in a more systemic context of transformation

units and systems, although with a marked theoretical

ﬂavour.

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