Fundamenta Informaticae XXI (2001) 1001–1032
1001
IOS Press
Lazy Graph Transformation
Fernando Orejas∗
Universitat Politècnica de Catalunya, Spain
[email protected]
Leen Lambers†
Hasso Plattner Institut, Universität Potsdam, Germany
[email protected]
Abstract. Applying an attributed graph transformation rule to a given object graph always implies
some kind of constraint solving. In many cases, the given constraints are almost trivial to solve. For
instance, this is the case when a rule describes a transformation G ⇒ H, where the attributes of H
are obtained by some simple computation from the attributes of G. However there are many other
cases where the constraints to solve may be not so trivial and, moreover, may have several answers.
This is the case, for instance, when the transformation process includes some kind of searching.
In the current approaches to attributed graph transformation these constraints must be completely
solved when defining the matching of the given transformation rule. This kind of early binding is
well-known from other areas of Computer Science to be inadequate. For instance, the solution chosen for the constraints associated to a given transformation step may be not fully adequate, meaning
that later, in the search for a better solution, we may need to backtrack this transformation step.
In this paper, based on our previous work on the use of symbolic graphs to deal with different aspects
related with attributed graphs, including attributed graph transformation, we present a new approach
that, based on the new notion of narrowing graph transformation rule, allows us to delay constraint
solving when doing attributed graph transformation, in a way that resembles lazy computation. For
this reason, we have called lazy this new kind of transformation. Moreover, we show that the approach is sound and complete with respect to standard attributed graph transformation. A running
example, where a graph transformation system describes some basic operations of a travel agency,
shows the practical interest of the approach.
Address for correspondence: Fernando Orejas, Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de
Catalunya, 08034 Barcelona, Spain
∗
The work of this author was partially supported by the MEC project FORMALISM (ref. TIN2007-66523) and by the AGAUR
grant to the research group ALBCOM (ref. 00516).
†
The work of this author was partially funded by the Deutsche Forschungsgemeinschaft in the course of the project - Correct
Model Transformations - see http://www.hpi.uni-potsdam.de/giese/projekte/kormoran.html?L=1.
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Keywords: Attributed graph transformation, symbolic graph transformation, lazy transformation
1.
Introduction
Attributed graphs and attributed graph transformation play a significant role in most applications of
graph transformation. In practice, an attributed graph transformation rule is like a normal rule, but some
nodes or edges are labelled by expressions over some given variables. Then, defining a match m of a
rule to a given object graph, whose attributes are some concrete values, means finding the values that
must be assigned to the variables occurring in the rule, so that the value of each expression associated
to each node or edge e on the left-hand side of the rule coincides with the value of the corresponding
attribute associated to m(e) in the object graph. That is, defining a match of a rule means solving
a set of constraints. In many cases, these constraints are trivial. For instance, when a rule describes
a transformation G ⇒ H, where the attributes in H are obtained by some simple computation from
the attributes in G, e.g. when the expressions used as attributes in the left-hand side of the rules are
just variables, and more general expressions, defined over these variables, only occur in the right-hand
side. However there are many other cases where the constraints to solve may be not so trivial and,
moreover, may have several answers. For instance, when the transformation process includes some kind
of searching (e.g. when the right-hand side of a rule involves a variable which does not occur explicitly
on the left-hand side). In existing approaches to attributed graph transformation these constraints must
be completely solved when defining the matching of the given transformation rule. Then, finding a
match means choosing one specific solution. This kind of early binding is well-known from other areas
of Computer Science to be inadequate. One problem is that the solution chosen for the constraints
associated to a given transformation step may not be fully adequate, meaning that we may need to
backtrack this transformation step. The approach taken in areas like Constraint Logic Programming [9],
by which our approach is inspired, is to postpone solving the constraints as much as possible, checking
meanwhile their satisfiability. Then, not only may we avoid some useless backtracking, but we have
other advantages. On the one hand, checking satisfiability may be computationally simpler than solving
a set of constraints, meaning that it may also be simpler to apply a transformation step. On the other
hand, some constraints which may be difficult to solve at a given moment, may become simpler, even
trivial, because of the interaction with constraints defined by later steps.
In [12], when studying the problem of defining graph constraints over attributed graphs, we saw that
the existing approaches [10, 1, 7, 5, 16] were not fully adequate for our purposes. These approaches
presented different kinds of technical difficulties together with a limited expressive power for defining
conditions on the attributes. To avoid these problems we presented a new formal approach which (we
believe) is conceptually simpler and more powerful than existing approaches. It is simple because, in our
approach, attributed graphs are not defined as some kind of combination of a graph and an algebra, as in
[10, 1, 7, 5], nor do we have to establish a difference between transformation rules and rule schemata,
as in [16]. At the same time, our approach is expressively more powerful, not only because we can
define graph constraints with arbitrary conditions on the attributes as we aimed, but also because we can
define transformation rules that cannot be defined in other approaches, as shown in [14]. Graphs in our
approach are called symbolic graphs, because the attributes in the graph are represented symbolically by
variables whose possible values are specified by a set of formulas.
However, in this paper, we show that symbolic transformation rules, and the corresponding notion of
Orejas, Lambers / Symbolic Graph Transformation
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symbolic graph transformation, as defined in [14], are too restrictive for the transformation of arbitrary
symbolic graphs. As a consequence, we introduce a more general notion of symbolic graph transformation rules, which we have called narrowing rules, and characterize graph transformation by means
of these rules. In addition, we study the compatibility of symbolic graph transformation with the semantics of symbolic graphs1 , showing that this kind of transformation is compatible with the semantics
of graphs, but not in a strong sense. Hence, we introduce a variant of symbolic graph transformation,
which we have called, narrowing graph transformation, proving that it is strongly compatible with respect to the semantics of graphs. Finally, using the notion of narrowing graph transformation, we present
a new approach to symbolic graph transformation that allows us to delay constraint solving when doing
attributed graph transformation. In particular we show that this new approach is sound and complete
with respect to standard attributed graph transformation, and it is complete and satisfies a property of
extended soundness with respect to symbolic graph transformation. Moreover, a running example, where
a graph transformation system describes some basic operations of a travel agency, shows the practical
interest of the approach.
The paper is organized as follows. In Section 2 we recapitulate some notions that are used in the
rest of the paper. In Section 3 we present the category and the semantics of symbolic graphs. Section 4
is dedicated to describe how (standard) symbolic graph transformation works. In Section 5 we present
the new notions of narrowing rules and narrowing graph transformation studying its compatibility with
respect to the semantics of symbolic graphs. Section 6 is dedicated to the new notion of lazy transformation and to prove its soundness and completeness. Finally, in Section 6, we compare our approach with
other related work and we draw some conclusions.
This paper is an extended version of [15], presented at ICGT 2010. In particular, in addition to the
detailed proofs of our results, we introduce the new notion of narrowing rules and narrowing transformation, presenting lazy transformation as a specific instance of narrowing transformation.
2.
Preliminaries
We assume that the reader has a basic knowledge on algebraic specification and on graph transformation.
For instance, we advise to look at [6] for more detail on algebraic specification or at [3] for more detail
on graph transformation.
2.1.
Basic algebraic concepts and notation
A signature Σ = (S, Ω) consists of a set of sorts S, and Ω is a family of operation and predicate
symbols typed over these sorts. A Σ-algebra A consists of an S-indexed family of sets {As }s∈S and
a function opA (resp. a relation prA ) for each operation op (resp. each predicate pr) in the signature.
A Σ-homomorphism h : A → A0 consists of an S-indexed family of functions {hs : As → A0s }s∈S
commuting with the operations and preserving the relations.
Given a signature Σ, we denote by TΣ the term algebra, consisting of all the possible Σ-(ground)
terms. Given any Σ-algebra A there is a unique homomorphism hA : TΣ → A. In particular, hA yields
the value of each term in A. Similarly, TΣ (X) denotes the algebra of all Σ-terms with variables in
X, and given a variable assignment σ : X → A, this assignment extends to a unique homomorphism
1
This notion is defined in Section 5.
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σ # : TΣ (X) → A yielding the value of each term after the replacement of each variable x by its value
σ(x). In particular, when an assignment is defined over the term algebra, i.e. σ : X → TΣ , then σ # (t)
denotes the term obtained by substituting each variable x in t by the term σ(x). However, for simplicity,
even if it is an abuse of notation, we will write σ(t) instead of σ # (t).
2.2.
E-graphs and Attributed Graphs
E-graphs are introduced in [3] as a first step to define attributed graphs. An E-graph is a kind of labelled
graph, where nodes and edges may be decorated with labels from a given set. The difference with labelled
graphs, as commonly understood, is that in labelled graphs it is usually assumed that each node or edge
is labelled with a given number of labels, which is fixed a priori. In the case of E-graphs, each node or
edge may have any arbitrary (finite) number of labels, which is not fixed a priori. Actually, in the context
of graph transformation, the application of a rule may change the number of labels of a node or of an
edge.
Formally, in E-graphs labels are considered as a special class of nodes and the labeling relation
between a node or an edge and a given label is represented by a special kind of edge. For instance, this
means that the labeling of an edge is represented by an edge whose source is an edge and whose target is
a label.
Definition 2.1. (E-Graphs and morphisms)
N L , E EL , s , sN L , sEL , t , tN L , tEL )
An E-graph over the set of labels XG is a tuple G = (VG , XG , EG , EG
G G
G G
G
G
G
consisting of:
• VG and XG , the sets of graph nodes and of label nodes, respectively.
N L , and E EL , the sets of graph edges, node label edges, and edge label edges, respectively.
• EG , EG
G
and the source and target functions:
• sG : EG → VG and tG : EG → VG
L
NL
NL
NL
• sN
G : EG → VG and tG : EG → XG
EL
EL
• sEL
G : EG → EG and tEL : EG → XG
An E-graph is linear if every label x is attached to at most one node or one edge, i.e. there is at most
one (node or edge) label edge e such that x = tjG (e), where j ∈ {N L, EL}.
Given the E-graphs G and G0 , an E-graph morphism f : G → G 0 is a tuple, hfV : VG → VG0 , fX :
0 ,f
NL
0N L
EL
0EL
XG → XG0 , fE : EG → EG
EN L : EG → EG , fEEL : EG → EG i such that f commutes with
all the source and target functions.
E-graphs and E-graph morphisms form the category E − Graphs.
The following constructions on E-graphs are needed in the sections below. The first one tells us how
to replace the labels of an E-graph, and is used to define the semantics of a symbolic graph (cf. Definition
3.3):
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Definition 2.2. (Label substitution)
N L , E EL , s , sN L , sEL , t , tN L , tEL ), a set of labels X’, and
Given an E-graph G = (VG , XG , EG , EG
G G
G G
G
G
G
a function h : XG → X 0 we define the graph resulting from the substitution of XG along h, h(G) =
0 , E 0N L , E 0EL , s , s0 N L , s0 EL , t0 , t0 N L , t0 EL ), where:
(VG0 , X 0 , EG
G
G
G
G
G
G
G
G
0 = E , E 0N L = E N L , E 0EL = E EL , s0 = s , s0 N L = sN L , s0 EL = sEL , and
• VG0 = VG , EG
G
G
G
G
G
G
G
G
G
G
G
t0G = tG
0N L : t0N L (e) = h(tN L (e))
• For every e ∈ EG
G
G
0EL : t0EL (e) = h(tEL (e))
• For every e ∈ EG
G
G
Moreover, h induces the definition of the morphism h : G → h(G), with h = hidV , h, idEG , idEN L , idEEL i.
Notice that if f : G → H is a morphism such that fV , fE , fEN L , and fEEL are bijections, then we
can consider that f is induced by the label substitution fX (up to isomorphism).
We have used the same notation for the morphism associated to a label substitution and the given
substitution. This is obviously an abuse of notation, but we believe that it introduces no confusion while
it simplifies notation.
Reach(G) eliminates all the labels in an E-graph which are not bound to a node or an edge and
Restr(G, Y ) eliminates all the labels which are not in Y , assuming that Y includes all reachable labels
(and, perhaps, some unreachable ones). These constructions are used to define lazy graph transformation
(cf. Definition 6.1).
Definition 2.3. (Reachable subgraph and restriction subgraph)
Given an E-graph G = (VG , XG , EG , EN L , EEL , {sj , tj }j∈{G,N L,EL} ), we define the reachable subgraph of G, Reach(G) as the largest subgraph of G where XReach(G) consists of all labels x such that:
• There is a node label edge nl ∈ EN L such that x = tN L (nl), or
• there is an edge label edge el ∈ EEL such that x = tEL (el).
Moreover, given a morphism f : G → G 0 we denote by Reach(f ) : Reach(G) → G0 the restriction
of f to the subgraph Reach(G).
Similarly, we define the restriction of G to a set of labels Y , Restr(G, Y ), where XReach(G) ⊆ Y as
the largest subgraph of G where XRestr(G,Y ) = Y .
Finally, let us define attributed graphs as presented in [3]. In particular, an attributed graph is an
E-graph whose labels are the values of a given data algebra that is assumed to be included in the graph.
Definition 2.4. (Attributed graphs and morphisms)
An attributed graph over Σ is a pair hG, Di, where D is a given Σ-algebra, called the data algebra of
the graph,
U and G is an E-graph such that the set XG of labels in G
U consists of all the values in D, i.e.
XG = s∈S Ds , where s is the set of sorts of the data algebra and denotes disjoint union.
Given the attributed graphs over Σ AG = hG, Di and AG0 = hG0 , D0 i, an attributed graph morphism h : AG → AG0 is a pair hhgraph , halg i, where hgraph is an E-graph morphism, hgraph : G → G0
and halg is a Σ-homomorphism, halg : D → D0 such that the values in D are mapped consistently by
hgraph and halg , i.e. for each sort s ∈ S the diagram below commutes:
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Ds _
halg
/ D0
s _
XG
hgraph
/ XG0
Attributed graphs and attributed graph morphisms form the category AttGraphs.
3.
Symbolic graphs
In this section, we provide a small introduction to symbolic graphs. In particular, in the first subsection,
we introduce the category of this kind of graphs and describe some results that can be found in [14]. Then,
in the second subsection, we study the semantics of symbolic graphs presenting some new technical
results which are used in the rest of the paper.
3.1.
The category of symbolic graphs
If we consider that an attributed graph is like an E-graph whose labels are values over a given data
algebra, a symbolic graph can be seen as the specification of a class of attributed graphs. In particular,
a symbolic graph consists of an E-graph G whose labels are variables, together with a set of formulas Φ
that constrain the possible values of these variables. We consider that a symbolic graph denotes the class
of all attributed graphs where the variables in the E-graph have been replaced by values that make Φ
true in the given data domain. For instance, the symbolic graph in Figure 1 specifies a class of attributed
graphs, including distances in the edges, that satisfy the well-known triangle inequality. We must note
that the graph in this figure is not really an E-graph, but a user-friendly representation of an E-graph.
In particular, we have not depicted the edge label edges that would bind the labels d1 , d2 and d3 to the
corresponding graph edges. Actually, in the rest of the paper we will always omit depicting (node or
edge) label edges, since we think that the resulting graphs are more intuitive.
d3
d1
d2
with d3 ! d1+d2
Figure 1.
A symbolic graph
Definition 3.1. (Symbolic graphs and morphisms)
A symbolic graph SG over the data Σ-algebra D, with Σ = (S, Ω), is a pair SG = hG, Φi, where G is
an E-graph over an S-sorted set of variables, used as labels, X = {Xs }s∈S , i.e. XG = ∪s∈S Xs , and Φ
is a set of first-order Σ-formulas with free variables in X and including elements in D as constants.
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Given symbolic graphs hG1 , Φ1 i and hG2 , Φ2 i over D, a symbolic graph morphism h : hG1 , Φ1 i →
hG2 , Φ2 i is an E-graph morphism h : G1 → G2 such that D |= Φ2 ⇒ h(Φ1 ), where h(Φ1 ) is the set of
formulas obtained when replacing in Φ1 every variable x1 in the set of labels of G1 by hX (x1 ).
Symbolic graphs over D together with their morphisms form the category SymbGraphsD .
As said in the definition, we consider that Φ is a set of arbitrary first-order Σ-formulas. However,
for practical purposes, we may want to restrict the class of formulas that can occur in this set, since
first-order formula satisfiability is an undecidable problem. In principle, the only condition that we
need to ensure that the results presented in this paper hold is that the given class of formulas is closed
under conjunction, disjunction and existential quantification, because these are the connectives needed
to ensure the existence of pushouts and pullbacks in the category of symbolic graphs, as proved in [14].
Remark 3.1. (Symbolic typed graphs)
Even if along the paper, for simplicity, we have only considered untyped graphs, in our running example
(see Example 3.3) we consider that our graphs are typed. We can easily extend all our theory to the case of
typed graphs using the standard technique. In particular, we just have to see typed graphs as morphisms
SG → T G, where SG is a symbolic graph and T G is the given type graph, which is a specific symbolic
graph (i.e. the category of typed symbolic graphs over the type graph T G is the slice category over T G).
In particular, type graphs T G must be symbolic graphs T G = hET G, Falsei, where ET G is an E-graph
over a set of labels XET G consisting of just one variable for each sort in the given data signature, and
False is the set consisting just of the f alse formula. In this way, given a symbolic graph SG = hG, Φi,
any (typing) morphism t : G → ET G for G is also a (typing) morphism t : SG → T G.
In [14], we showed that symbolic graphs are an adhesive HLR category taking as M-morphisms all
injective graph morphisms where the formulas constraining the source and target graphs are equivalent
(in most cases they will just be the same formula).
Definition 3.2. (M-morphisms)
An M-morphism h : hG, Φi → hG0 , Φ0 i is a monomorphism such that XG ∼
= XG0 , i.e. hX is a bijection,
0
and D |= h(Φ) ⇔ Φ
We will not study in detail all the constructions and results that are needed to prove that SymbGraphsD
is an adhesive HLR category (the interested reader is addressed to [14]). However, it is important to see
how pushouts work in order to understand symbolic graph transformation:
Proposition 3.1. [14] Diagram (1) below is a pushout if and only if diagram (2) is a pushout in E − Graphs
and D |= Φ3 ⇔ (g1 (Φ1 ) ∪ g2 (Φ2 )).
hG0 , Φ0 i
h2
hG2 , Φ2 i
h1
(1)
g2
/ hG1 , Φ1 i
g1
/ hG3 , Φ3 i
Therefore, as said above, we have:
Theorem 3.1. [14] SymbGraphsD is adhesive HLR.
G0
h2
G2
h1
(2)
g2
/ G1
g1
/ G3
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One may wonder whether SymbGraphsD is an adhesive category, in particular if it is adhesive
HLR when M-morphisms are just monomorphisms. The answer is no, as it can be seen from the counterexample below:
Example 3.1. Let us consider the diagram below, where all the arrows are the identity graph morphism.
Then, it is easy to see that the bottom diagram is a pushout in SymbGraphsD , but it is not a van
Kampen square [11].
hG, {true}i
hG, {true}i
'
s
hG, {false}i
'
s
hG, {false}i
hG, {false}i
hG, {true}i
'
s
hG, {false}i
'
hG, {false}i
s
In particular, we may easily check that all the morphisms in the diagram are monomorphisms in
SymbGraphsD . Moreover, we can also check easily that the top and bottom squares are pushouts
and the back squares are pullbacks2 . However the front-left square is not a pullback. In particular,
true 6= false ∨ false.
Moreover, we may see that, in general, for diagrams along monomorphisms, pushout complements
are not unique as the following counterexample shows:
Example 3.2. Let us consider the morphisms f : hG, {true}i → hG, {false}i and g : hG, {false}i →
hG, {false}i, where both morphisms are identities as E-graph morphisms, then it is easy to see that the
diagrams below are pushouts:
hG, {true}i
hG, {false}i
3.2.
f
(1)
/ hG, {false}i
hG, {true}i
g
/ hG, {false}i
hG, {true}i
f
(2)
/ hG, {false}i
g
/ hG, {false}i
Semantics of symbolic graphs
As we said above, we may consider that a symbolic graph SG denotes a class of attributed graphs. In
particular the class of all attributed graphs that can be obtained by replacing the variables in SG by values
2
If the given formulas do not include variables, pullbacks in SymbGraphsD are defined in terms of pullbacks in
E − Graphs, where the associated formula of the resulting graph is the disjunction of the formulas of the graphs involved.
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in the given data algebra, so that the associated condition becomes true in the algebra:
Definition 3.3. (Semantics of symbolic graphs)
The semantics of a symbolic graph hG, Φi over a data algebra D is a class of attributed graphs defined:
Sem(hG, Φi) = {hσ(G), Di | σ : XG → D and D |= σ(Φ)}
where σ(G) denotes the graph obtained according to Def. 2.2.
We may notice that the class of attributed graphs denoted by a symbolic graph may be empty if the
associated condition is unsatisfiable.
Every attributed graph may be seen as a symbolic graph by just replacing all its values by variables
and by including an equation xv = v, for each value v in the data algebra, into the corresponding set of
formulas, where xv is the variable that has replaced the value v. We call these kind of symbolic graphs
grounded symbolic graphs. For instance, in Figure 2, on the right, we can see the symbolic representation
of the attributed graph on the left. However, to enhance readability, in our figures we will often show
values of the given data algebra as attributes of the given graph, instead of replacing them by variables
and showing the corresponding equation. This is done, for instance, in Example 3.3, where a grounded
symbolic graph is displayed as an attributed graph.
18
d3
15
12
d2
d1
with d1= 12, d2= 15, d3= 18
Figure 2.
Attributed graph and grounded symbolic graph
Definition 3.4. (Grounded symbolic graphs)
A symbolic graph hG, Φi over a data algebra D is grounded
if
-11. XG includes a variable, which we denote by xv , for each value v ∈ D, and
2. For every substitution σ : XG → D, such that D |= σ(Φ), we have σ(xv ) = v, for each variable
xv ∈ XG .
Moreover, we define GSymbGraphsD as the full subcategory of SymbGraphsD consisting of
all grounded graphs.
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It should be obvious that the semantics of a grounded graph includes exactly one attributed graph,
and that grounded graphs are closed up to isomorphism. Moreover, in [14] we can see that for every
attributed graph AG there is a unique grounded symbolic graph (up to isomorphism) GSG(AG) such
that Sem(GSG(AG)) consists of AG. In particular, the E-graph associated to GSG(AG) is obtained
substituting every data value v in a set of labels by a variable xv , and the set of formulas in the symbolic
graph consists of an equation xv = v, for each value v in D.
In [14] we also showed that, given a data algebra D, the category of attributed graphs over D, as
presented in [3], and the subcategory of grounded symbolic graphs over D are equivalent, provided that
the given data algebra is finitely generated. As a consequence, we may identify them. For instance, we
may consider that the semantics of a symbolic graph is a class of grounded symbolic graphs.
We may notice that, if h : SG → SG0 is a symbolic graph morphism and SG0 is grounded then
the condition D |= Φ0 ⇒ h(Φ), that the symbolic morphism h must satisfy, can be represented as a
constraint satisfaction problem:
Fact 3.1. Given a symbolic graph SG = hG, Φi and a grounded symbolic graph SG0 = hG0 , Φ0 i, an
E-graph morphism h : G → G0 is a symbolic graph morphism if and only if D |= h0 (Φ), where
h0 : XG → D is the mapping defined ∀x ∈ XG : h0 (x) = v if h(x) = xv .
Proof:
By definition, we know that h is a symbolic graph morphism if and only if D |= Φ0 ⇒ h(Φ). This is
equivalent to showing that, for every substitution σ, if D |= σ(Φ0 ) then D |= σ(h(Φ)). Now, Φ0 consists
of all the equations xv = v, therefore the only substitution σ such that D |= σ(Φ0 ) is σ(xv ) = v. But for
every variable x in XG , h0 (x) = σ ◦ h(x). Therefore, this is equivalent to D |= h0 (Φ).
t
u
The above fact means that defining a morphism from SG = hG, Φi into a grounded symbolic graph
can be seen partially as a constraint satisfaction problem. In particular, the morphism must provide values
to the variables in Φ so that the conditions (constraints) in Φ are satisfied. For instance, in Example 4.2
we see an object graph that represents a customer of a travel agency that wants to make a hotel reservation
for two days in Berlin and has a certain budget. Then, in that example, applying a transformation rule
that is supposed to implement this reservation (i.e. defining the match morphism) implies finding a hotel
a to be bound to a variable h of sort hotel that satisfies two conditions: that a is located in Berlin and
that its price for two nights is cheaper than the given budget.
Let us now show two facts that are needed in the rest of the paper. The first one shows that for any
symbolic graph SG and the grounded representation of any attributed graph AG in the semantics of SG
there exists a symbolic morphism between SG and GSG(AG), which is induced by a label substitution.
Fact 3.2. An attributed graph AG = hG, Di is in the semantics of SG = hG0 , Φi if and only if there is
a symbolic graph morphism h : SG → GSG(AG) induced by a label substitution.
Proof:
If AG ∈ Sem(SG) this means that G = σ(G0 ) for a given variable substitution σ such that D |= σ(Φ).
But this means that we may define h as the morphism associated to σ 0 ◦ σ, where σ 0 is the substitution
that maps every value v ∈ D to the variable xv . In addition, D |= σ(Φ) implies, by Fact 3.1, that h is a
symbolic graph morphism. Conversely, if there is a symbolic morphism h : SG → GSG(AG) induced
by a given label substitution, we can define an assignment σ : XG0 → D with σ(x) = v if h(x) = x0
Orejas, Lambers / Symbolic Graph Transformation
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and the equation x0 = v is in ΦGSG(AG) . Then, by Fact 3.1, we know that D |= σ(Φ) implying that the
attributed graph hσ(G0 ), Di is in Sem(SG).
t
u
The second fact states that, if AG is in the semantics of SG0 and the variables in SG are a subset of
the variables of SG0 then any morphism f from a symbolic graph SG to GSG(AG) can be factorized
into two morphisms f 0 : SG → SG0 and h : SG0 → GSG(AG), where h is the morphism defined in
the previous fact.
Fact 3.3. Given symbolic graphs SG = hG, Φi and SG0 = hG0 , Φ0 i, such that G is linear (cf. Definition
2.1), and given a morphism f : SG → GSG(AG), where AG ∈ Sem(SG0 ), there is an E-graph
morphism f 0 : SG → SG0 such that f = h ◦ f 0 , where h : SG0 → GSG(AG) is the morphism defined
in Fact 3.2.
Proof:
It is enough to define f 0 as follows:
• For every graph node or edge e, f 0 (e) = h−1 (f (e)).
• For every label x ∈ XG , f 0 (x) = x0 , if x is the target in G of some node label edge or of some
edge label edge e (i.e. x is attached to a node or an edge via e) and the target of f 0 (e) in G0 is x0 .
• For every label x ∈ XG , f 0 (x) = x0 ,if x is not attached to any node or edge in G and x0 is any
label in G0 such that f (x) = h(x0 ).
It is routine to check that f 0 is an E-morphism because, on the one hand, if h is the morphism induced
by a label substitution, h is a bijection on graph nodes and edges, and on the other hand, G is linear, which
means that the definition of f 0 on the labels is correct. Moreover, by definition f = h ◦ f 0 .
t
u
Example 3.3. (Introduction to Running Example)
In our running example we specify a travel agency in terms of typed symbolic graphs and symbolic graph
transformation rules. In particular, the data signature of our example is:
Sorts int, bool, city, hotel, flight
Opns price : hotel → int
location : hotel → city
price : flight → int
departure : flight → city
destination : flight → city
and the data algebra, depicted by means of two tables in Figure 3.
The type graph, depicted in Figure 4, consists of a Customer that can request a hotel reservation
HotelRequest or a flight reservation FlightRequest. These requests can be responded by a FlightReservation or a HotelReservation, respectively. A Customer has the labels name and bud, describing the
total budget that the customer is willing to spend for his reservations. A HotelRequest has as labels the
number of nights and loc, describing the city where the hotel is located. A FlightRequest has as labels
1012
Orejas, Lambers / Symbolic Graph Transformation
hotel
price
city
h1
h2
h3
h4
h5
100
90
85
70
200
München
Berlin
Berlin
Köln
Berlin
flight
price
departure
destination
f1
f2
f3
f4
175
115
150
85
Berlin
Berlin
Barcelona
Barcelona
Barcelona
Barcelona
Berlin
Berlin
«uses»
book flight
«uses»
book travel
book hotel
Figure 3.
A data algebra
«uses»
book transfer
Traveler
Customer
-name : string
-bud : int
int, bool, city, hote
price:hotel->int
location: hotel ->
price:flight->int
departure: flight destination: flight
FlightRequest
HotelRequest
-dep : city
-dest : city
-loc : city
-nights : int
HotelReservation
FlightReservation
-h : hotel
-f : flight
Figure 4.
Type graph
fr : the
FlightRequest
reserveFlight
fr : FlightRequest
dep and dest, the departure and destination city of
flight, respectively.
The HotelReservation
and
dep
:
city
dep : city
FlightReservation have only one label h and f for the hotel and flight that are reserved, respectively.
dest : city
dest : city
: FlightRe
f : flight
Finally, a possible start graph describing the needs of a customer called Alice is depicted in Figure
3 and shows a customer with a budget of 450,
c : Customer
5. In particular, this graph is assumed to be grounded
c : Customer
name
string
requesting a hotel for two nights in Berlin, a flight from :Barcelona
to Berlin and a flightname
from: string
Berlin to
bud : int
bud : int
Barcelona.
dep = departure(f) && dest = destination(f) && bud’ = bud - price(f) &&
3
As explained above, for readability, it is depicted as an attributed graph
hr : HotelRequest
loc : city
nights : int
c : Customer
reserveHotel
hr : HotelRequest
loc : city
nights : int
: HotelRese
h : hotel
0,-./1*8*,0/
0,-./1*8*,0/
9*8*?31/(;%"
.*8*.(/%&
9*8*?31/(;%"
0#;%*8*1/",0'36*8*,0/
0#;%*8*1/",0'36@*8*,0/
815$%$815&(/10*9,$--$2)!3$%$2)!$: *#'/5"*9,$;$0/<9(.,$--$2)!36%7
Orejas, Lambers / Symbolic Graph Transformation
1/#"/*-"#7.8
8*A(/%&=%>3%1/
8*<&,-./=%>3%1/
&(9*8*9,/:*D*E%"&,0
0,-./1*8*,0/*D*F
6%7*8*9,/:*D*E#"9%&(0#
6%1/*8*9,/:*D*E%"&,0
8*?31/(;%"
8*<&,-./=%>3%1/
0#;%*8*1/",0-*D*G&,9%
'36*8*,0/*D*HIJ
6%7*8*9,/:*D*E%"&,0
6%1/*8*9,/:*D*E#"9%&(0#
Figure 5.
4.
1013
Start graph
Symbolic graph transformation for attributed graph transformation
In the previous section we have seen that the category of symbolic graphs is adhesive HLR. Hence,
following [3], we can define a first notion of symbolic graph transformation using spans of M-morphisms.
This is equivalent to consider that the left and right-hand sides (and also the interface) of a rule are
constrained by the same set of formulas. This means that we may denote symbolic graph transformation
rules as pairs hL ←- K ,→ R, Φi, where L, K and R are E-graphs over the same set of labels X and Φ
is a set of formulas over X and over the values in D. Intuitively, Φ relates the attributes in the left and
right-hand sides of the rule and it may also impose some constraints on the matchings. Moreover, for
technical reasons we require that L is linear (cf. Definition 2.1), meaning that no label is bound to two
different elements in L4 .
Definition 4.1. (Symbolic graph transformation rule)
A symbolic graph transformation rule is a pair hL ←- K ,→ R, Φi, where L, K, R are E-graphs over
the same set of labels X, L is linear, L ←- K ,→ R is a span of E-graph inclusions5 , and Φ is a set of
formulas over X and over the values in the given data algebra D.
Example 4.1. (Symbolic graph transformation rules)
In Figure 6 we can see the rules that describe some operations of the travel agency of our example.
In particular, rule reserveFlight connects a FlightReservation with a FlightRequest for some Customer.
Identical names in rules specify that the corresponding nodes are preserved. Edges between nodes with
identical names are preserved as well. The rule includes a formula Φ, expressing that the departure city
dep and destination city dest of the FlightRequest should be equal to the departure city departure(f)
and destination city destination(f) of the reserved flight f, respectively. Moreover the total budget bud
of the customer is diminished by the price of the reserved flight price(f) and as an extra constraint it is
required that the new budget bud’ is still greater than or equal to zero. The rule reserveHotel connects
a HotelReservation with a HotelRequest for some Customer. It holds a formula Φ, expressing that the
location loc of the HotelRequest should be identical to the location of the reserved hotel location(h). The
total budget bud of the customer is diminished by the price of the reserved hotel multiplied with the
4
If needed we can transform SL into an equivalent linear graph (with the same semantics). We just need to change the repeated
occurrences of the same variable by new variables and add the corresponding equalities to the given set of formulas.
5
or, in general, monomorphisms
Customer
flight
-name : string
-bud : int
int, bool, city, hotel, flight
price:hotel->int
location: hotel -> city
-dep : city
price:flight->int
-dest : city
departure: flight -> city
Orejas, Lambers / Symbolic Graph Transformation
destination: flight -> city
-price : string
-source
-destination
FlightRequest
HotelRequest
-loc : city
-nights : int
1014
HotelReservation
hotel
-price : string
-location
FlightReservation
city
number
-h : hotel
of nights price(h)*nights
and as an extra constraint it is required again that the new budget bud’
-f : flight
is still greater than or equal to zero.
Notice that the fact that these rules are spans of M-morphisms implies that the variable f in the first
Customer
rule must be considered to be included in the left-hand side of the rule, even if it is not bound to any node
-name : string
or edge of the left-hand side graph. The same happens with the variable h in the second rule, as well-bud'
as : int
with bud’ in both rules.
fr : FlightRequest
reserveFlight
dep : city
dest : city
fr : FlightRequest
: FlightReservation
dep : city
dest : city
f : flight
c : Customer
c : Customer
name : string
bud : int
name : string
bud' : int
dep = departure(f) && dest = destination(f) && bud’ = bud - price(f) && bud’ >=0
hr : HotelRequest
reserveHotel
loc : city
nights : int
hr : HotelRequest
: HotelReservation
loc : city
nights : int
h : hotel
c : Customer
c : Customer
name : string
bud : int
name : string
bud' : int
loc = location(h) && bud’ = bud – (price(h) * nights) && bud’>=0
start graph:
Figure 6.
: FlightRequest
: HotelRequest
loc : city
nights : int
Graph transformation rules
dep : city
dest : city
: FlightRequest
Customer
As usual, :the
application of a graph
rule hL ←- K ,→ R, Φi to a given symbolic
dep :transformation
city
name : string
dest
:
city
graph SG can
be defined by a double pushout in the category of symbolic graphs:
bud : int
Definition 4.2. (Symbolic graph transformation)
Given symbolic graphs SG and SH over a Σ-algebra D, a symbolic graph transformation rule, r =
hL ←- K ,→ R, Φr i over D, and a morphism m : hL, Φr i → SG, we say that SH is a direct transformation of SG by r via m, denoted SG =⇒r,m SH, if SH can be obtained by the following double
pushout in SymbGraphsD :
hL, Φr i o
m
SG o
? _ hK, Φr i ? _ SF


/ hR, Φr i
m0
/ SH
1015
Orejas, Lambers / Symbolic Graph Transformation
Moreover, we write SG =⇒ SH if the given rule and match can be left implicit, and we write =⇒∗
to denote the reflexive and transitive closure of =⇒.
The proposition below states some properties that are important to understand symbolic graph transformation. The first property states that the set of labels and the condition of the given object graph
remain invariant after symbolic transformation. The second property states that symbolic transformation
preserves groundedness. The third property shows that the application of a symbolic transformation rule
can be defined in terms of a transformation of E-graphs. Finally, the last condition states that if we can
apply a transformation to a given graph SG, then we can also apply (essentially) the same transformation
to a graph that extends SG with some additional variables and whose condition is more restrictive.
Proposition 4.1.
Given a symbolic graph transformation rule r = hL ←- K ,→ R, Φi over a given data algebra D:
1. If hG, Φ0 i =⇒r,m hH, Φ00 i then XG = XH and D |= Φ0 ⇔ Φ00 .
2. If hG, Φ0 i =⇒r,m hH, Φ00 i and hG, Φ0 i is grounded then hH, Φ00 i is also grounded.
3. If m : L → G is an E-graph morphism, we have that hG, Φ0 i =⇒r,m hH, Φ0 i if and only if the
diagram below is a double pushout in E − Graphs and D |= Φ0 ⇒ m(Φ).
Lo
m
Go
? _K 
/R
? _F 
/H
4. If hG, Φ0 i =⇒r,m hH, Φ0 i, XG ⊆ X, and D |= Φ00 ⇒ Φ0 then hG∪X, Φ00 i =⇒r,i◦m hH ∪X, Φ00 i,
where G ∪ X denotes the graph obtained by adding to G all the labels that are in X but not in XG
and i is the inclusion i : G ,→ G ∪ X.
Proof:
1. According to the definition of symbolic graph transformation rules, XL = XK = XR implying
that the morphisms in the span hL, Φi ←- hK, Φi ,→ hR, Φi are M-morphisms. If hG, Φ0 i =⇒r,m
hH, Φ00 i, this means that the diagram below is a double pushout in SymbGraphsD :
hL, Φi o
m
hG, Φ0 i o
? _ hK, Φi 
m00
? _ hF, Ψi

/ hR, Φi
m0
/ hH, Φ00 i
But we know [14] that pushouts in SymbGraphsD preserve M-morphisms (see [14]), hence the
morphisms in the span hG, Φ0 i ←- hF, Ψi ,→ hH, Φ00 i are also M-morphisms. This means that
XG = XF = XH and D |= Φ0 ⇔ Ψ ⇔ Φ00 .
1016
Orejas, Lambers / Symbolic Graph Transformation
2. A direct consequence of the previous item. The reason is that, if Φ0 defines the values of all the
variables in XG , then Φ00 also defines the values of all the variables in XH = XG .
3. If hG, Φ0 i =⇒r,m hH, Φ0 i this means that m is a symbolic graph morphism and the diagram below
is a double pushout in SymbGraphsD :
? _ hK, Φi 
hL, Φi o
m
(1)
m00
? _ hF, Φ0 i hG, Φ0 i o
/ hR, Φi
(2)

m0
/ hH, Φ0 i
Now, according to Proposition 3.1, this means that the diagram below is a double pushout in
E − Graphs. Moreover, since m is a symbolic graph morphism then D |= Φ0 ⇒ m(Φ).
? _K 
Lo
m
Go
(3)
m00
? _F 
/R
m0
(4)
/H
Conversely, let us suppose that diagrams (3) and (4) are pushouts in E − Graphs. We know
that if the morphisms in the span L ←- K ,→ R are the identity on variables then so are (up
to isomorphism) the morphisms on the span G ←- F ,→ H. This means that the morphisms
m, m0 and m00 coincide on the variables and, as a consequence, if D |= Φ0 ⇒ m(Φ), then
D |= Φ0 ⇒ m0 (Φ) and D |= Φ0 ⇒ m00 (Φ). Hence all the morphisms in (1) + (2) are symbolic
graphs morphisms. Therefore, to prove that (1) and (2) are pushouts, according to Prop. 3.1, we
have to prove that D |= Φ0 ⇔ Φ0 ∪ m(Φ) and D |= Φ0 ⇔ Φ0 ∪ m0 (Φ), but this is obvious if
D |= Φ0 ⇒ m(Φ) and D |= Φ0 ⇒ m0 (Φ).
4. If hG, Φ0 i =⇒r,m hH, Φ0 i this means that D |= Φ0 ⇒ m0 (Φ) and H can be obtained by the double
pushout below:
? _K 
/R
Lo
m
Go
m00
? _F 
m0
/H
We know that i◦m satisfies the gluing conditions with respect to r, since i just adds some variables
that are not connected to any node or edge. Therefore, we can build the following double pushout
diagram:
Lo
i◦m
G∪X o
? _K 
/R
? _F ∪ X 
/H ∪X
But, if D |= Φ00 ⇒ Φ0 and i(m0 (Φ)) = m0 (Φ), since i is the equality on all labels in XG , then
D |= Φ00 ⇒ i(m0 (Φ)) implying hG ∪ X, Φ00 i =⇒r,i◦m hH ∪ X, Φ00 i.
1017
Orejas, Lambers / Symbolic Graph Transformation
t
u
The fact that the formula constraining the variables of the E-graph H after transformation is the
same formula for the graph G before transformation does not mean that the attributes in the graph do not
change after transformation. The reason is that the transformation may change the bindings of variables
to nodes and edges, as the example below shows. Moreover, according to Fact 3.1, if we apply the rule
hL ←- K ,→ R, Φi to a grounded symbolic graph SG = hG, Φ0 i, the condition D |= Φ0 ⇒ m(Φ) can
be seen as the constraint satisfaction problem where we find the values to assign to the variables in Φ, so
that this set of formulas is satisfied in D.
Example 4.2. In Figure 7 we can see how the rule reserveHotel presented in Example 4.1 is applied to
a given start graph as presented in Example 3.3, describing a customer that would like to reserve a hotel
for 2 days in Berlin and, in addition, would also like to reserve a flight from Barcelona to Berlin and
return.
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
reserveHotel
: HotelReservation
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
h : hotel = h5
: Customer
: FlightRequest
name : string = Alice
bud : int = 450
dep : city = Berlin
dest : city = Barcelona
: Customer
: FlightRequest
name : string = Alice
bud : int = 50
dep : city = Berlin
dest : city = Barcelona
Berlin = location(h) && bud = 450 –– (price(h) * 2) && bud >=0
Figure 7.
reserveFlight
: HotelReservation
h : hotel
Symbolic transformation
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
fr1 : FlightReservation
f : flight
: Customer
: FlightRequest
name : string = Alice
bud : int
dep : city = Berlin
dest : city = Barcelona
In this transformation, a hotel in Berlin is chosen non-deterministically. This choice is made when
defining the matching between the left-hand side of the rule and the given graph. The reason is that, as
Berlin = location(h) && Barcelona = departure(fr1.f) && Berlin = destination(fr1.f) && bud = 450 ––
said above, even
if it*is2) not
explicitly
depicted,
the left-hand side of ReserveHotel is supposed to include
(price(h)
- price(fr1.f)
&& bud
>=0
the variable h, because that variable is explicitly in the right-hand side of the rule and hence it should
: HotelRequest
: FlightRequest
fr1 : FlightReservation
also
be included in all the graphs
the dep
rule.
when defining
the match morphism we must bind
reserveFlight
loc : city =in
Berlin
: cityThen,
= Barcelona
f : flight
nights
:
int
=
2
dest
:
city
=
Berlin
that variable to
some
hotel
hi
in
D
such
that
the
formula
hotel
: HotelReservation
h : hotel
: FlightRequest
: Customer
fr2 : FlightReservation
f : flight
dep : city = Berlin
Berlin = location(hi)name
&&: string
bud’=450-(price(hi)*2)
&& bud’>=0
= Alice
dest : city = Barcelona
bud : int
Berlin =the
location(h)
Barcelona
= departure(fr1.f)
&& Berlin
= destination(fr1.f)
&& Berlin
= . This kind of
is satisfied. Here,
chosen&&
hotel,
h5, satisfies
the above
formula
for h=h5 and
bud’=50
departure(fr2.f) && Barcelona = destination(fr2.f) && bud = 450 –– (price(h) * 2) - price(fr1.f) early bindingprice(fr2.f)
may not&&bebud
very
efficient. For instance, after the transformation defined above no flight
>=0
reservation is possible, because the budget of the user would not be sufficient anymore for reserving any
flight connecting Barcelona and Berlin.
Hence, we would need to backtrack this transformation and to
hotel price location
try a different
hotel.
This
situation
also
occurs
h1
100 when
Münchenworking with attributed graphs and attributed graph
int, bool, city, hotel, flight
h2
90
Berlin
transformation rules as in [3].
price:hotel->int
h3
85
Berlin
As shown
in hotel
[14], ->
symbolic
graphh4transformation
is more powerful than attributed graph transforma70
Köln
location:
city
tion as presented
in
[3].
In
particular,
any
attributed
graph
transformation rule r can be represented by
price:flight->int
flight price departure
destination
departure:
flight -> city
a symbolic
graph transformation
rulef1SR(r)
such
that an attributed
graph AG can be transformed by r
175
Berlin
Barcelona
destination: flight -> city
f2
115
Berlin
Barcelona
f3
150
Barcelona
Berlin
f4
85
Barcelona
Berlin
Possible solutions in the form (h,fr1.f,fr2.f,bud):
1018
Orejas, Lambers / Symbolic Graph Transformation
into a graph AH if and only if the grounded graph associated to AG, GSG(AG), can be transformed by
SR(r) into GSG(AH). This representation is quite obvious. If ar = AL ←- AK ,→ AR is an attributed
graph transformation rule, usually this means that AL, AR and AK are graphs over the term algebra, i.e.
they include terms (with variables) as labels. This means that we can, straightforwardly, represent r as a
symbolic graph transformation rule SR(r) replacing the graphs involved by their corresponding associated grounded graphs. Note that, in this case, the conditions in the rule (which would coincide with the
name
conditions in the grounded
Customergraphs GSG(AL), GSG(AR), and GSG(AK)) would be equations Xt = t,
string
where t is a term-name
occurring
as an attribute in the rule. In this context, as saidCustomer
above, it is proven that, given
: string
attributed graphs-bud
AG,
AH, AG =⇒r,m AH if and only if GSG(AG) =⇒bud
: int
SR(r),GSG(m) GSG(AH).
int rules that cannot be
However, the converse is not true. In [14], we also proved that there are symbolic
HotelRequest
FlightRequest
simulated by attributed graph transformation rules.
nights
HotelRequest
-loc : city
-nights : int
5.
FlightRequest
dep
loc
-dep : city
-dest : city
dest
Narrowing rules and narrowing graph transformation
city
hotel powerful flight
As we have seen above, symbolic graph transformation can be hconsidered
enough to fspecify
transformations of attributed graphs as studied in [3]. However,
the kind of rules that we have
studied in
HotelReservation
FlightReservation
HotelReservation
FlightReservation
Section
4 are not powerful enough
to specify transformations of arbitrary symbolic graphs that may be
-h : hotel
-f : flight
considered
reasonable. For instance,
given the start graph G in Figure 8, we would be unable to apply
the rule ReserveHotel to G. The problem is that we cannot match the variable h, which (as said above) is
implicitly in the left hand side of the rule, to any variable in G. We could think that a possible solution is
to assume that all symbolic graphs implicitly include a denumerable set of variables of each sort, so that
h can be matched to some variable in G of sort hotel6 . However, this is not enough. If f is supposed to
be the match morphism then the condition in G (i.e. true) must imply f (Φ), where Φ is the condition
of the rule. However, in general, this is obviously false, since true only implies tautologies.
Customer
-name : string
-bud : int
Figure 8.
HotelRequest
-loc : city
-nights : int
Start graph
The problem is that a rule consisting of a span of M-morphisms can transform the structure of an
object graph or change the binding of attributes to nodes or edges of the object graph, but it cannot
add new information to the graph, where this new information is represented by new variables and new
conditions over the variables. To overcome this problem, we consider a more general kind of rules, which
we call narrowing transformation rules, because, as we may see below, when applied to a symbolic
graph, it restricts or narrows their associated conditions by adding the conditions of the rule. These rules
consist of a span of monomorphisms, where only the left morphism is assumed to be an M-morphism.
6
We may note that, in Example 4.2, the start graph does implicitly include a denumerable set of variables, even if they are not
depicted in Figure 5. The reason is that, formally, that graph is assumed to be grounded, which means that it includes a variable
per each existing data value.
1019
Orejas, Lambers / Symbolic Graph Transformation
Definition 5.1. (Narrowing graph transformation rule)
A narrowing graph transformation rule over a Σ-algebra D is a triple hΦL , L ←- K ,→ R, ΦR i, where
L, K are E-graphs over the same set of labels, L is linear, L ←- K ,→ R is a span of E-graph inclusions7 ,
which means that XL = XK ⊆ XR , ΦL and ΦR are sets of Σ-formulas over XL and XR , respectively,
and over the values in D, and finally D |= ΦR ⇒ ΦL .
Note that the fact that the morphism hK, ΦL i ,→ hL, ΦL i is an M-morphism guarantees that the
pushout complement in a rule application is unique, if it exists (see Example 3.2). This means that we
may define symbolic graph transformation, using narrowing rules, by double pushouts, in the standard
way:
Definition 5.2. (Symbolic graph transformation with narrowing rules)
Given symbolic graphs SG and SH over a Σ-algebra D, a narrowing rule, r = hΦL , L ←- K ,→ R, ΦR i
over D, and a morphism m : hL, ΦL i → SG, we say that SH is a direct transformation of SG by r via
m, denoted SG =⇒r,m SH, if SH can be obtained by the double pushout in SymbGraphsD :
? _ hK, ΦL i 
hL, ΦL i o
m
(1)
/ hR, ΦR i
(2)
? _ SF 
SG o
m0
/ SH
and Sem(SH) is not empty. As usual, we write SG =⇒ SH if the given rule and match can be left
implicit, and we write =⇒∗ to denote the reflexive and transitive closure of =⇒.
We may notice that, in principle, the conditions ΦSH of the symbolic graph SH obtained by the
above double pushout if hG, Φi =⇒r,m hH, ΦSH i, could be unsatisfiable in D, which means that the
semantics of hH, Φ∪m0 (ΦR )i would be empty. Obviously, it makes little sense to make a transformation
whose result is inconsistent. For this reason, this is forbidden in this definition. We may also notice that,
if SG =⇒r,m SH, SG is a grounded graph and r is a narrowing rule then SH may be not grounded,
because the conditions added by the application of r may be satisfied by several non-isomorphic graphs.
As in the case of transformations by (standard) symbolic rules, symbolic graph transformations can
be defined in terms of E-graph transformations. As in Proposition 4.1, this is just a consequence of how
pushouts in SymbGraphsD are defined.
Fact 5.1. Given a narrowing transformation rule r = hΦL , L ←- K ,→ R, ΦR i over D and an E-graph
morphism m : L → G, we have that hG, Φi =⇒r,m hH, Φ ∪ m0 (ΦR )i if and only if the diagram below
is a double pushout in E − Graphs and D |= Φ ⇒ m(ΦL ).
? _K 
Lo
m
Go
(1)
? _F 
/R
(2)
m0
/H
Remark 5.1. Obviously, narrowing rules are a generalization of symbolic graph transformation rules,
in the sense that every symbolic rule r = hL ←- K ,→ R, Φr i can be seen as the narrowing rule
7
or, in general, monomorphisms
1020
Orejas, Lambers / Symbolic Graph Transformation
narrow(r) = hΦr , L ←- K ,→ R, Φr i. Then, as a consequence of the definition of graph transformation, SG =⇒r,m SH if and only if SG =⇒narrow(r),m SH.
The fact that symbolic graph transformation with narrowing rules can be defined in terms of E-graph
transformation means that it is easy to extend the fundamental results that characterize graph transformation (e.g. the Embedding and Extension theorems or the Church-Rosser and Parallelism Theorems)
to this kind of transformations.
As said above, symbolic graphs can be seen as specifications of classes of attributed graphs or, equivalently, of grounded symbolic graphs. Therefore symbolic graph transformations, with standard rules or
with narrowing rules can be seen as transformations of specifications. In this context, one may ask if
graph transformation as defined above is compatible with respect to the semantics of symbolic graphs,
in the sense that the semantics of a transformation coincides, in some sense, with the corresponding
transformation of the semantics. More precisely:
Definition 5.3. (Compatibility of symbolic transformation and semantics)
Symbolic graph transformation is compatible with the semantics of symbolic graphs if for every narrowing transformation rule r = hΦL , L ←- K ,→ R, ΦR i, every transformation SG =⇒r,m SH,
and every AH ∈ Sem(SH), there is an AG ∈ Sem(SG), such that GSG(AG) =⇒r,h◦m SH 0 and
AH ∈ Sem(SH 0 ), where h : SG → GSG(AG) is the morphism defined by Fact 3.2.
Moreover, symbolic graph transformation is strongly compatible with the semantics of symbolic
graphs if, in addition, for every AG ∈ Sem(SG), if GSG(AG) =⇒r,f SH 0 and f is factorized into
h◦m, with h : SG → GSG(AG) and m : L → G, according to Fact 3.3, there is a direct transformation
SG =⇒r,m SH such that Sem(SH 0 ) ⊆ Sem(SH).
Notice that, in the definition of strong compatibility, Fact 3.3, only ensures that m is an E-graph
morphism. Therefore, to ensure strong compatibility we would need to prove first that m is a symbolic
graph morphism. Then, we have:
Theorem 5.1. Symbolic graph transformation with narrowing rules is compatible, but not strongly compatible, with the semantics of symbolic graphs.
Proof:
• Compatibility: Let SG = hG, ΦSG i, SH = hH, ΦSH i, AH ∈ Sem(SH), SH 0 = GSG(AH) =
hH 0 , ΦSH 0 i, r = hΦL , L ←- K ,→ R, ΦR i, and suppose that SG =⇒r,m SH. This means that
diagrams (1) and (2) below are pushouts in the category of E-graphs, ΦSH = ΦSG ∪ m0 (ΦR ) and,
according to Fact 3.2 there is a morphism h0 : SH → SH 0 such that h0 is induced by a label substitution and D |= ΦSH 0 ⇒ h0 (ΦSH ). Now, we know that XL = XK ⊆ XR , therefore, without
loss of generality, we may assume that XG = XF ⊆ XH . Then, we can define a substitution σ
over the variables in G as σ(x) = h0 (x). Let G0 = σ(G), let F 0 = σ(F ), and let h : G → G0
and f : F → F 0 be the corresponding morphisms induced by σ. Then diagrams (3) and (4) below
are obviously pushouts, where the monomorphisms F 0 ,→ G0 and F 0 ,→ H 0 are defined like the
monomorphisms F ,→ G and F ,→ H on graph nodes and edges and they are the identity on the
variables.
Orejas, Lambers / Symbolic Graph Transformation
? _K 
Lo
m
(1)
h
(3)
G0 o
/R
m0
(2)

? _F Go
1021
/H
f
(4)
? _F 0 
h0
/ H0
This means that (1) + (3) and (2) + (4) are pushouts, and thus hG0 , Φ0 i =⇒r,h◦m hH 0 , Φ0 ∪ h0 ◦
0 , (x = v) ∈ Φ
m0 (ΦR )i, where Φ0 = {x = v | x ∈ XG
SH 0 }. Notice that this definition of
0
0
Φ is correct since all the variables in G are in H 0 , and SH 0 is a grounded model, so for each
variable x in H 0 there should be an equation x = v in ΦSH 0 . Finally, we can easily see that
AH ∈ Sem(hH 0 , Φ0 ∪ h0 ◦ m0 (ΦR )). It is enough to notice that, by construction, we can obtain
AH replacing in H 0 each variable x by the value v, if (x = v) ∈ ΦSH 0 .
• Strong compatibility: To prove that symbolic graph transformation with narrowing rules is not
complete in the above sense, it is enough to consider the following counter example. Let SG be a
symbolic graph over the integers data algebra with only one node labelled with a variable x, and
with the empty set of conditions. And consider a rule that says that if we have a graph consisting
of a node labelled with a variable y whose value is 2 (i.e. ΦL consists of the equation y = 2) then
we can replace y by z whose value is 3. Now, we cannot apply that rule to SG with the obvious
matching, since the empty set of conditions does not imply x = 2. However, we can apply the rule
to the grounded graph that includes the equation x = 2.
t
u
In this context, the solution to obtain a strongly compatible transformation relation is quite obvious.
If the problem is caused by not allowing to apply a rule if the conditions of the object graph do not imply
the conditions of the left-hand side of the rule, the solution is to allow this application independently of
these conditions. We can call this kind of transformation narrowing transformations, and we can show
(see Theorem 5.2) that they are strongly compatible with respect to the semantics of symbolic graphs.
Definition 5.4. (Narrowing graph transformation)
Given a symbolic graph SG = hG, ΦSG i over a Σ-algebra D, a narrowing transformation rule r =
hΦL , L ←- K ,→ R, ΦR i over D, and an E-graph morphism m : L → G, we define the direct narrowing
transformation of SG by r via m, denoted SG Vr,m SH, as the symbolic graph SH = hH, ΦSH i, such
that Sem(SH) 6= ∅, where H is obtained by the double pushout in E − Graphs below:
? _K Lo
m
Go
(1)
? _F 

/R
(2)
m0
/H
and where ΦSH = ΦSG ∪ m0 (ΦR )
We may notice that, when applying narrowing transformations, the left-hand side condition of a rule,
ΦL , plays no role in the transformations. This means that we may also apply narrowing transformation
1022
Orejas, Lambers / Symbolic Graph Transformation
using standard rules. In particular, narrowing transformation with a (standard) symbolic rule r = hL ←K ,→ R, Φr i is equivalent to symbolic transformation with the narrowing rule r = h∅, L ←- K ,→
R, Φr i.
We can see that narrowing graph transformation is more general than symbolic graph transformation
with narrowing rules:
Fact 5.2. Given r = hΦL , L ←- K ,→ R, ΦR i, if SG =⇒r,m SH then SG Vr,m SH.
Proof:
According to Fact 5.1, if SG =⇒r,m SH, where SG = hG, ΦSG i and r = hΦL , L ←- K ,→ R, ΦR i,
then SH = hH, ΦSG ∪ m0 (ΦR )i, where H is obtained by the double pushout diagram below:
Lo
m
Go
? _K 
/R
? _F 
m0
/H
But, by definition of narrowing graph transformation, this means SG Vr,m SH.
t
u
The converse, obviously, is not true in general. To end this section, we can show the strong compatibility of narrowing transformation with respect to the semantics of symbolic graphs:
Theorem 5.2. Narrowing graph transformation is strongly compatible with the semantics of symbolic
graphs.
Proof:
• Compatibility: The proof is the same as the proof of compatibility in Theorem 5.1
• Strong compatibility: If AG ∈ Sem(SG), with SG = hG, ΦSG i, and GSG(AG) Vr,h◦m SH 0 ,
where GSG(AG) = hG0 , ΦGSG(AG) i, h : SG → GSG(AG) is the morphism defined by Fact
3.2, and m : L → G, this means that SH 0 = hH 0 , ΦGSG(AG) ∪ m0 (ΦR )i, where H 0 is obtained by
the double pushout below:
? _K 
Lo
h◦m
(1)
G0 o
? _F 0 
/R
(2)
m0
/ H0
but, given the match m : L → G, we can also apply the transformation SG Vr,m SH:
? _K 
Lo
m
Go
(3)

? _F /R
(4)
m00
/H
where SH = hH, ΦSG ∪ m00 (ΦR )i. Now, pushouts (1) and (2) can be decomposed:
Orejas, Lambers / Symbolic Graph Transformation
? _K 
Lo
m
(3)
Go
h
G0 o
(5)

? _F ? _F 0 
1023
/R
m00
(4)
/H
(6)
h0
/ H0
where m0 = h0 ◦ m00 . On the one hand, we know that XL ⊆ XR , this means that XG ⊆ XH
and XG0 ⊆ XH . On the other hand, we also know that h is a morphism induced by a label
substitution, implying that it is a bijection on graph nodes and edges. This means that h0 is also a
bijection on graph nodes and edges and, therefore it is also induced by a label substitution. Hence,
as a consequence of Fact 3.2, if AH ∈ Sem(SH 0 ) we have that AH ∈ Sem(SH).
t
u
6.
Lazy graph transformation
In Example 4.2 we saw that, when defining a match for a rule r = hL ←- K ,→ R, Φr i, this match
must also be defined on variables that are bound to a node or edge in R but not bound to any node or
edge in L (i.e. intuitively, these variables are not part of L). This forces us to bind these variables too
early, implying that we may need to backtrack afterwards, as the example showed. The basic idea of
our approach to avoid this is that, when applying a rule, we do not need to match these variables. We
do this using two ideas. First, instead of applying (standard) symbolic graph transformation with the
given rule, we apply narrowing graph transformation. In this way, we do not ask the given match to
satisfy the conditions of the rule, but we only add these conditions afterwards. So, in a sense, we are
postponing the satisfaction of the associated constraints. However, this is not enough, because the match
morphism still has to bind the variables in L that are not bound to any node or edge. Then, the second
idea is that the match morphism for r should not be defined on L, but on Reach(L) (cf. Definition
2.3). This is equivalent to transforming each rule into a narrowing rule, whose E-graph in the left-hand
side is not L but Reach(L) and whose left condition is empty. As a consequence, when we apply the
associated narrowing rule to a graph SG, as we saw in the previous section, we add the formulas in Φr
to the result of the transformation, since these formulas define the possible values of the non-matched
variables with respect to the values of the rest of the variables, as we can see in Example 6.2. Moreover,
as we can also see in this example, after applying a lazy transformation we can simplify the resulting
condition. In particular, formula simplification can be considered correct because two symbolic graphs
are isomorphic when their corresponding E-graphs are identical (or isomorphic) and their conditions are
equivalent. In particular, applying some simplifications after a transformation may be interesting from a
practical viewpoint, since we may operate with the associated conditions in a more efficient way.
Definition 6.1. (Lazy graph transformation)
Given a transformation rule r = hL ←- K ,→ R, Φr i we define its associated lazy rule as the narrowing
rule Lazy(r) = h∅, Reach(L) ←- Restr(K, XReach(L) ) ,→ R, Φr i. Then, given a symbolic graph
SG = hG, ΦG i and a morphism m : Reach(L) → G, we define the lazy transformation of SG by r via
l
m, denoted SG =⇒r,m SH as the narrowing transformation SG VLazy(r),m SH.
1024
Orejas, Lambers / Symbolic Graph Transformation
l
This means that SG =⇒r,m SH if SH = hH, ΦH i, where H is obtained by the double pushout
below in the category of E-graphs:
? _ Restr(K, XReach(L) ) 
Reach(L) o
m
Go
(1)
/R
(2)
? _F 
m0
/H
l
and where ΦH = ΦG ∪ m0 (Φr ). As usual, we write SG =⇒ SH if the given rule and match can be left
l ∗
l
implicit, and we write =⇒ to denote the reflexive and transitive closure of =⇒.
Intuitively, H is similar to the E-graph obtained by a standard symbolic transformation, except that
it would include all the variables in XR \ XReach(L) instead of their corresponding matchings.
It may be noted that the resulting condition ΦH must be satisfiable for the given data algebra D. The
reason is that we have required that the result of a narrowing graph transformation must have non-empty
semantics.
Example 6.1. For example, if we apply the rule ReserveHotel as in Example 4.2 without binding the
variable h to some concrete value hi in Dhotel , then we obtain a lazy graph transformation step as depicted
in the first step of the lazy graph transformation in Example 6.2. After this first step the formula
Berlin = location(h) && bud=450-(price(h) * 2) && bud>=0
should be satisfied. Given the data algebra D as presented in Example 3.3, it is obvious that this formula
is satisfiable.
Let us now study the relation between symbolic graph transformation and lazy transformation for the
same rule. In particular, we are interested in studying if every symbolic transformation can be implemented in terms of a lazy transformation and, vice versa, if every lazy transformation corresponds to a
symbolic transformation. More precisely, we study this issue with respect to the semantics of specifications. These properties are formulated as follows:
l ∗
1. Soundness For every lazy transformation SG =⇒ SH and every attributed graph AH ∈ Sem(SH)
there is a symbolic transformation SG =⇒∗ SH 0 such that AH ∈ Sem(SH 0 ).
2. Completeness For every symbolic transformation SG =⇒∗ SH there is a lazy transformation
l ∗
SG =⇒ SH 0 such that Sem(SH) ⊆ Sem(SH 0 ).
Notice that, when the given SG is a grounded graph, the above properties ensure that we can get
exactly the same results with both kinds of transformation, although, in the case of lazy transformation
we may need to perform some constraint solving after the transformation process, as we will see in the
example below.
The first result tells us that, when SH is an arbitrary symbolic graph, lazy graph transformation is
a complete implementation of symbolic graph transformation, but it may be considered, technically, not
sound. In particular, it may be possible to apply a lazy transformation to a graph SG = hG, ΦG i using
the rule r = hL ←- K ,→ R, Φr i, but it may be impossible to apply the same rule to SG to do symbolic
1025
Orejas, Lambers / Symbolic Graph Transformation
transformation. The reason is that for symbolic transformation we require that D |= ΦG ⇒ m(Φr ), for a
given match m, while in the case of lazy transformation it may be enough that ΦG ∪ m(Φr ) is satisfiable
in D. This may happen, for instance, if the condition on G is 0 < X < 5 and the condition on the rule is
3 < X < 7. In that case, we would be unable to apply symbolic graph transformation, since 0 < X < 5
does not imply 3 < X < 7. However, it could be possible to apply lazy transformation and, then, the
resulting condition would be 0 < X < 5 ∧ 3 < X < 7, or equivalently 3 < X < 5. Instead, we will
show that lazy transformation satisfies a different property, which we call extended soundness.
Theorem 6.1. (Extended soundness and completeness of lazy graph transformation)
Given a symbolic graph SG = hG, ΦG i, then:
l ∗
1. Extended soundness For every lazy transformation SG =⇒ SH, with SG = hG, ΦSG i and
SH = hH, ΦSH i there is a symbolic transformation hG ∪ XH , ΦSH i =⇒∗ SH (up to isomorphism), where G ∪ XH denotes the graph obtained by adding to G all the variables in H (actually
the variables that are in H but not in G, since XG ⊆ XH ).
2. Completeness For every symbolic transformation SG =⇒∗ SH there is a lazy transformation
l ∗
SG =⇒ SH 0 such that Sem(SH) ⊆ Sem(SH 0 ).
Proof:
l ∗
1. Extended Soundness Given the lazy transformation SG =⇒ SH, with SG = hG, ΦSG i and
SH = hH, ΦSH i we prove that there is a symbolic transformation hG ∪ XH , ΦSH i =⇒∗ SH
by induction on the length of the derivation. If SG = SH, then the case is trivial. For the
l ∗
l
general case, let us suppose that SG =⇒ SG1 and SG1 =⇒r,m SH, with SG1 = hG1 , ΦSG1 i,
SH = hH, ΦSH i where H is defined by the diagram below:
? _ Restr(K, XReach(L) ) 
Reach(L) o
m
G1 o
(1)
? _ F1 
/R
(2)
m0
/H
and ΦSH = ΦSG1 ∪ m0 (Φr ).
By induction, we may assume that there is a symbolic transformation hG∪XG1 , ΦSG1 i =⇒∗ SG1 .
By Proposition 4.1.4, this means that hG ∪ XH , ΦSH i =⇒∗ hG1 ∪ XH , ΦSH i, since XG ∪ XG1 ⊆
XH and D |= ΦSH ⇒ ΦSG1 . Therefore, we have to prove that there is a match m1 : SL →
hG1 ∪XH , ΦSH i such that hG1 ∪XH , ΦSH i =⇒r,m1 SH. Now, without loss of generality, we may
assume that XG1 = XF1 ⊆ XH and that m0 is the equality for every variable in XR \ XReach(L)
(if needed we could rename the variables in r). Then, XH = XG1 + (XR \ XReach(L) ) = XG1 +
(XL \ XReach(L) ). As a consequence, we can define the match m1 : hL, Φr i → hG1 ∪ XH , ΦSH i
as follows. For any graph node or any edge e ∈ L: m1 (e) = m(e); for every label x ∈ XReach(L) :
m1 (x) = m(x); and for every x ∈ (XL \ XReach(L) ): m1 (x) = x. Then, m1 is a symbolic
morphism since D |= ΦSH ⇒ m1 (Φr ), since ΦSH = ΦSG1 ∪ m0 (Φr ) and m0 (Φr ) = m(Φr ) =
1026
Orejas, Lambers / Symbolic Graph Transformation
m1 (Φr ). Hence, we have the direct transformation hG1 ∪ XH , ΦSH i =⇒r,m1 hH 0 , ΦSH i, where
H 0 is defined by the following double pushout diagram:
? _K 
Lo
m1
(3)
/R
(4)
G1 ∪ XH o
? _F 0 
m01
/ H0
1
Now, we know that L = Reach(L) + (XL \ XReach(L) ) and K = Restr(K, XReach(L) ) + (XL \
XReach(L) ). Moreover, G1 ∪ XH = G1 + (XL \ XReach(L) ), because we have assumed that m0 is
the equality for every variable in (XR \ XReach(L) ) and, hence, XH = XG1 + (XR \ XReach(L) ).
Therefore, F10 = F1 + (XL \ XReach(L) ). But then pushout (2) can be decomposed in the two
pushouts below:
Restr(K, XReach(L) ) 
/ K 
/ F0 
1

F1 /R
(4)
m01
/ H0
and this means that H = H 0 (up to isomorphism), implying hG1 ∪ XH , ΦSH i =⇒r,m1 SH.
2. Completeness We proceed by induction on the length of the transformation. In particular we will
l ∗
prove that if SG =⇒∗ SH, with SH = hH, ΦSH i, then SG =⇒ SH 0 , with SH 0 = hH 0 , ΦSH 0 i,
and there is a label substitution h : XH 0 → XH such that h(H 0 ) = H and D |= h(ΦSH 0 ) ⇔ ΦSH .
We may notice that this implies that if AH ∈ Sem(SH) then AH ∈ Sem(SH 0 ). The reason is
that if AH ∈ Sem(SH) this means that there is a substitution σ such that AH = hσ(H), Di
and D |= σ(ΦSH ), but this implies that AH = hσ(h(H 0 )), Di and D |= σ(h(ΦSH 0 )) and, so,
AH ∈ Sem(SH 0 ).
The base case, i.e. SG = SH, is trivial. For the general case, let us assume that SG =⇒∗
SG1 and SG1 =⇒r,m SH, with SG1 = hG1 , ΦSG1 i and SH = hH, ΦSH i, with ΦSG1 =
ΦSH , since graph conditions remain invariant after symbolic graph transformation with symbolic
transformation rules (see Proposition 4.1.3) and where H is defined by the double pushout below:
? _K 
Lo
m
G1 o
(1)
? _F 
/R
(2)
j
m0
/H
l ∗
By induction, we may assume that there is a lazy transformation SG =⇒ SG01 with SG01 =
hG01 , ΦSG01 i and a label substitution h1 : XG01 → XG1 such that h1 (G01 ) = G1 and D |=
h1 (ΦSG01 ) ⇔ ΦSG1 . Now, we define the match morphism m1 : Reach(L) → G01 as follows.
For any graph node or any edge e ∈ Reach(L):
1027
Orejas, Lambers / Symbolic Graph Transformation
• For any graph node or any edge e ∈ Reach(L): m1 (e) = h−1
1 (m(e))
• For every label x ∈ XReach(L) , where x is attached to some node or edge by a node label
edge or an edge label edge e (i.e. e ∈ EN L ∪ EEL and tL (e) = x)8 : m1 (x) = y if y is the
target of m1 (e) in G01 .
This definition is correct because, on the one hand, h1 is a bijection on graph nodes and edges and,
on the other hand, because L is a linear symbolic graph and therefore each variable is attached to at
most one node or edge. We may notice that, as a consequence of the commuting properties of target
functions with graph morphisms, h1 ◦ m1 = m ◦ i, where i is the inclusion i : Reach(L) ,→ L.
Now let us consider the following diagrams, where all the squares are pushouts (m◦i, and therefore
h ◦ m1 , satisfy the gluing conditions because m and i satisfy them):
? _ Restr(K, XReach(L) )
_
o
Reach(L)

_
? _K
Lo
m
m1
(3)
i
G1 o
(1)
? _ Restr(K, XReach(L) )
Reach(L) o
h1
? _F
? _F 0
G01 o
m00
1
(4)
(5)
h00
? _ F 00
G1 o
Now, since the composition of pushouts is a pushout, we know that (1) + (3) and (4) + (5)
are pushouts. Moreover, since pushout complements are unique in M-adhesive categories and
h1 ◦ m1 = m ◦ i then F = F 00 . Thus, h00 is a morphism from F 0 to F . Moreover, since h is a
morphism induced by label substitution, so is h00 . Let us now consider the diagram below:
Restr(K, XReach(L) ) /R
m00
1
(6)
h00
j0
(7)

0
F

F

j
m01
/ H0
h0
/H
where (6) is a pushout, j is defined in diagram (2) above, and h0 is the unique morphism that satisfies h0 ◦ j 0 = j ◦ h00 and h0 ◦ m01 = m0 , by the universal property of pushout (6). Please, notice that
diagram (7) is not necessarily a pushout. Let us now see that h0 is a morphism induced by a label
substitution. Without loss of generality we may assume that H 0 = F 0 +(R\Restr(K, XReach(L) ))
and H = F + (R \ K), where + denotes disjoint union. Then, if e is a graph node or edge from
F 0 , we know that h0 (j 0 (e)) = j(h00 (e)), but h00 is a morphism induced by a label substitution and
assuming, without loss of generality, that j and j 0 are inclusions then we have h0 (e) = e. Similarly,
if e is a graph node or edge in R \ Restr(K, XReach(L) ), then h0 (m01 (e)) = m0 (e), and assuming
8
L
tL is assumed to denote the corresponding target function in L, i.e. tN
or tEL
L
L .
1028
Orejas, Lambers / Symbolic Graph Transformation
again without loss of generality that m0 and m01 are the identity on the elements in R \ K and
R \ Restr(K, XReach(L) ), respectively, we have h0 (e) = e.
Hence, to end the proof, we just have to show that D |= h0 (ΦSH 0 ) ⇔ ΦSH . We know that ΦSH 0 =
(ΦSG01 ∪ m01 (Φr )), D |= ΦSH ⇒ m(Φr ), m(Φr ) = m0 (Φr ), and D |= h1 (ΦSG01 ) ⇔ ΦSH .
Therefore, we have h0 (ΦSH 0 ) = h0 (ΦSG01 ∪ m01 (Φr )) = h0 (ΦSG01 ) ∪ h0 (m01 (Φr )) = h1 (ΦSG01 ) ∪
m0 (Φr ), because h0 (ΦSG01 ) = h1 (ΦSG01 ) and h0 ◦ m01 = m0 . This implies D |= h0 (ΦSH 0 ) ⇔
ΦSH ∪ m0 (Φr ). But, since D |= ΦSH ⇒ m0 (Φr ), we have D |= h0 (ΦSH 0 ) ⇔ ΦSH .
t
u
The second result tells us that if the given graph SG is grounded (or, equivalently, it is an attributed
graph) then lazy transformation is sound and complete with respect to symbolic graph transformation.
This means that, semantically, it does not matter if, each time we apply a transformation rule, we solve
the constraints associated to that application, or if we apply lazy transformation and, at the end, we solve
the constraints of the resulting graph.
Theorem 6.2. (Soundness and completeness of lazy graph transformation over grounded graphs)
Given a grounded symbolic graph SG = hG, ΦG i, then:
l ∗
1. Soundness For every lazy transformation SG =⇒ SH and every attributed graph AH ∈ Sem(SH)
there is a symbolic transformation SG =⇒∗ SH 0 such that AH ∈ Sem(SH 0 ).
2. Completeness For every symbolic transformation SG =⇒∗ SH there is a lazy transformation
l ∗
SG =⇒ SH 0 such that Sem(SH) ⊆ Sem(SH 0 ).
Proof:
1. Soundness We proceed by induction on the length of the derivation. In particular we will prove
l ∗
that if SG =⇒ SH, with SH = hH, ΦSH i, and σ : XH → D satisfies D |= σ(ΦSH ), then
there is a transformation SG =⇒∗ SH 0 , a label substitution h : XH → XG , and an assignment
σ 0 : XG → D, such that SH 0 = hH 0 , ΦSG i is grounded, h(SH) = SH 0 , and σ = σ 0 ◦ h. This is
obviously enough to ensure that AH = hσ 0 (H 0 ), Di is in Sem(SH 0 ). Notice that, asking for the
condition h(SH) = SH 0 , when h : XH → XG , makes sense because XG = XH 0 , since symbolic
graph transformation preserves the variables of graphs.
l ∗
If SG = SH, the case is trivial. For the general case, let us suppose that SG =⇒ SG1 and
l
SG1 =⇒r,m SH, with SG1 = hG1 , ΦSG1 i, SH = hH, ΦSH i where H is defined by the double
pushout diagram below:
? _ Restr(K, XReach(L) ) Reach(L) o
m
G1 o
(1)
? _ F1 

/R
(2)
m0
/H
and ΦSH = ΦSG1 ∪ m0 (Φr ), and let σ : XH → D be an assignment such that D |= σ(ΦSH ).
Without loss of generality, we may assume that XH = XG1 + (XR \ XReach(L) ) and that for
Orejas, Lambers / Symbolic Graph Transformation
1029
every x ∈ XReach(L) : m(x) = m0 (x). Now, let σ1 be the restriction of σ to XG1 . Obviously,
D |= σ1 (ΦSG1 ) .
By induction, we may assume that there is a symbolic transformation SG =⇒∗ SG01 , a label
substitution h1 : XG1 → XG , and an assignment σ 0 : XG → D, such that SG01 = hG01 , ΦSG i is
grounded, h1 (SG1 ) = SG01 , and σ1 = σ 0 ◦ h1 .
Let m1 : L → G01 be the following match. For every element e ∈ Reach(L) (e is a graph node or
edge) m1 (e) = h1 (m(e)) and for every variable x ∈ XL , m1 (x) = xv , where v is the value in D
such that σ(m0 (x)) = v. Notice that this definition is correct because, on the one hand, we have
assumed that for every x ∈ XReach(L) : m(x) = m0 (x), and on the other hand, G01 is grounded,
so it includes a variable xv . Now, we can see that m1 : hL, Φr i → SG01 is a symbolic graph
morphism, i.e. that D |= ΦSG01 ⇒ m1 (Φr ). We know that SG01 is grounded, therefore σ 0 is the
only substitution such that D |= σ 0 (ΦSG01 ) and this means that σ 0 (xv ) = v, for every value v. By
definition of m1 , we have that, for every variable x ∈ XL = XR , σ 0 (m1 (x)) = σ(m0 (x)) and we
know that D |= σ(ΦSH ) and D |= ΦSH ⇒ m0 (Φ(r)). But this means D |= σ(m0 (Φ(r))) and,
hence, D |= σ 0 (m1 (Φr )). In addition, m1 satisfies the gluing conditions with respect to the rule r
because m satisfies them by assumption, and h1 satisfies them because it is a morphism associated
to a label substitution. Therefore, we can apply the symbolic transformation SG01 =⇒r,m1 SH 0
and we can define the label substitution h : XH → XG as follows. For every x ∈ XH , if x ∈ XG1
then h(x) = h1 (x), and if x ∈ XL \ XReach(L) then h(x) = m1 (x). Now, SH is grounded
because symbolic graph transformation preserves groundedness. Moreover, it is routine to check
that h(SH) = SH 0 , and σ = σ 0 ◦ h.
2. Completeness This is just a special case of Theorem 6.1.2.
t
u
Example 6.2. Consider the lazy graph transformation as presented in Figure 9, where a hotel is reserved
for two nights in Berlin and a flight from Barcelona to Berlin and return is reserved. The labels h for
the hotel, fr1.f and fr2.f for the flights are not bound to a concrete hotel or concrete flights yet. Note that
we use the *.* notation to distinguish labels having the same name in different nodes. At the end of this
transformation a formula equivalent to:
Berlin = location(h) && Barcelona = departure(fr1.f) &&
Berlin = destination(fr1.f) && Berlin = departure(fr2.f) &&
Barcelona = destination(fr2.f) && bud = 450 - (price(h) * 2)
- price(fr1.f) - price(fr2.f) && bud >=0
should be satisfied9 . Solving this formula for the data algebra D given in Example 3.3, we obtain the
following solutions for the labels (h,fr1.f,fr2.f,bud): (h2,f4,f1,10), (h3,f4,f1,20), (h2,f3,f2,5), (h3,f3,f2,15),
(h2,f4,f2,70) and (h3,f4,f2,80). In particular, for the solution (h2,f4,f1,10) a corresponding symbolic
transformation exists, where h, fr1.f and fr2.f are bound to h2, f4 and f1, respectively. Moreover, in the
end the total budget bud of the customer is 10.
9
Actually,the formula that we would obtain is a bit more complex including several (renamed copies of the variable bud, since
each application of a rule would add a new variable bud and a constraint defining it. However, if we would allow to simplify
the given formula after each transformation step, as it is sometimes done in Constraint Logic Programming, we could indeed
obtain the above formula
1030
Orejas, Lambers / Symbolic Graph Transformation
To solve the resulting constraints after a sequence of transformations we may need to use a specialized constraint solver. If that solver gives the possibility of defining some kind of optimality criteria (e.g.
the overall price of the reservations) then it would return us the best solution according to these criteria.
However, if the data algebra D can be considered to be implemented by a database, as it may happen in
this example, then the resulting constraints after a sequence of transformations could be compiled into a
query to that database, meaning that all the possible solutions would just be obtained as a result of that
query.
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
reserveHotel
: HotelReservation
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
h : hotel
: Customer
: FlightRequest
name : string = Alice
bud : int = 450
dep : city = Berlin
dest : city = Barcelona
: Customer
: FlightRequest
name : string = Alice
bud : int
dep : city = Berlin
dest : city = Barcelona
Berlin = location(h) && bud = 450 –– (price(h) * 2) && bud >=0
reserveFlight
: HotelReservation
h : hotel
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
fr1 : FlightReservation
f : flight
: Customer
: FlightRequest
name : string = Alice
bud : int
dep : city = Berlin
dest : city = Barcelona
Berlin = location(h) && Barcelona = departure(fr1.f) && Berlin = destination(fr1.f) && bud = 450 ––
(price(h) * 2) - price(fr1.f) && bud >=0
reserveFlight
: HotelReservation
h : hotel
: HotelRequest
: FlightRequest
loc : city = Berlin
nights : int = 2
dep : city = Barcelona
dest : city = Berlin
fr1 : FlightReservation
f : flight
: FlightRequest
: Customer
dep : city = Berlin
dest : city = Barcelona
name : string = Alice
bud : int
fr2 : FlightReservation
f : flight
Berlin = location(h) && Barcelona = departure(fr1.f) && Berlin = destination(fr1.f) && Berlin =
departure(fr2.f) && Barcelona = destination(fr2.f) && bud = 450 –– (price(h) * 2) - price(fr1.f) price(fr2.f) && bud >=0
Figure
Lazy
graph transformation
hotel9. price
location
int, bool, city, hotel, flight
7.
h1
100
München
h2
90
Berlin
h3
85
Berlin
price:hotel->int
h4
70
location: hotel -> city
price:flight->int
flight price
departure: flight
-> city
Conclusion
and
relatedf1work
175
destination: flight -> city
f2
115
Köln
departure
destination
Berlin
Barcelona
Berlin
Barcelona
f3
150
Barcelona graph
Berlin
In this paper we have studied several forms
of symbolic
transformation, including showing that we
f4
85
Barcelona
Berlin
can delay constraint solving in the framework of attributed graph transformation by means of symbolic
graphs. The idea of the technique comes from constraint logic programming [9]. Actually, the idea unin the form (h,fr1.f,fr2.f,bud):
derlying Possible
symbolicsolutions
graphs also
comes from constraint logic programming, where the domain of values,
(h2,f4,f1,440),(h3,f4,f1,430),(h2,f3,f2,445),(h3,f3,f2,435),(h2,f4,f2,380),(h3,f4,f2,370)
and the conditions on them are neatly separated from symbolic clauses. In a similar way, we separate
Wrong solutions: (h2,f3,f1,505),(h3,f3,f1,495),(h5,fr1.f,fr2.f,bud)
Optimal solution (bud is minimal): (h3,f4,f2,370)
Remark: could this optimal solution have been computed without backtracking, using
additional axioms? In each step, check that there is no other value in the algebra
Orejas, Lambers / Symbolic Graph Transformation
1031
the graph part in symbolic graphs from the conditions defining the value part. Symbolic graph transformation can be seen as a generalization of attribute grammars [8]. In particular, attribute grammars can
be considered to provide a method for computing attributes in trees. Moreover, these attributes must be
computed in a fixed way: either top-down (inherited attributes) or bottom-up (synthesized attributes).
There are three kinds of approaches to define attributed graphs and attributed graph transformation.
In [10, 1] attributed graphs are seen as algebras. In particular, the graph part of an attributed graph is
encoded as an algebra extending the given data algebra. In [7, 5] an attributed graph is a pair (G, D)
consisting of a graph G and a data algebra D whose values are nodes in G. In [2] these two approaches
are compared showing that they are, up to a certain point, equivalent. Finally, [16] is based on the use
of labelled graphs to represent attributed graphs, and of rule schemata to define graph transformations
involving computations on the labels. That approach has some similarities with our approach, including
the simplicity provided by the separation of the algebra part and the graph part of attributed graphs.
However, the conceptual difference between the notions of attributed graph and of rule schemata has
the problem that this approach does not fit as an instance of adhesive rule-based transformation, which
means that we cannot apply to this approach the rich theory of adhesive categories. Moreover, we do not
think that there is a direct way of defining lazy graph transformations for any of those approaches, since
there is no clear way of handling constraints explicitly.
There are further aspects related to symbolic graph transformations that deserve some future work.
Addressing the implementation of these techniques could be an obvious goal. However, there are other
fundamental aspects which may be interesting to study, like developing specific techniques for checking
confluence for symbolic transformation or the definition of symbolic graph transformation with borrowing contexts, avoiding some existing technical difficulties. In particular, when computing the borrowed
context of a given rule application, we are forced to assign values of the given data algebra to all the variables in the left hand-side of the given rule. This causes that, in general, for any possible rule application
we may have an infinite number of contexts that only differ on the values of the attributes. Obviously, this
problem poses difficulties for using the borrowing context technique to prove bisimilarity for attributed
graph processes.
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