Lazy Graph Transformation

Transcription

1 Fundaenta Inforaticae XXI (2001) IOS Press Lazy raph Transforation Fernando Orejas Universitat Politècnica de Catalunya, Spain Leen Labers Hasso Plattner Institut, Universität Potsda, erany Abstract. Applying an attributed graph transforation rule to a given object graph always iplies soe kind of constraint solving. In any cases, the given constraints are alost trivial to solve. For instance, this is the case when a rule describes a transforation H, where the attributes of H are obtained by soe siple coputation fro the attributes of. However there are any other cases where the constraints to solve ay be not so trivial and, oreover, ay have several answers. This is the case, for instance, when the transforation process includes soe kind of searching. In the current approaches to attributed graph transforation these constraints ust be copletely solved when defining the atching of the given transforation rule. This kind of early binding is well-known fro other areas of Coputer Science to be inadequate. For instance, the solution chosen for the constraints associated to a given transforation step ay be not fully adequate, eaning that later, in the search for a better solution, we ay need to backtrack this transforation step. In this paper, based on our previous work on the use of sybolic graphs to deal with different aspects related with attributed graphs, including attributed graph transforation, we present a new approach that, based on the new notion of narrowing graph transforation rule, allows us to delay constraint solving when doing attributed graph transforation, in a way that resebles lazy coputation. For this reason, we have called lazy this new kind of transforation. Moreover, we show that the approach is sound and coplete with respect to standard attributed graph transforation. A running exaple, where a graph transforation syste describes soe basic operations of a travel agency, shows the practical interest of the approach. Address for correspondence: Fernando Orejas, Departaent de Llenguatges i Sistees Inforàtics, Universitat Politècnica de Catalunya, Barcelona, Spain The work of this author was partially supported by the MEC project FOMALISM (ref. TIN ) and by the AAU grant to the research group ALBCOM (ref ). The work of this author was partially funded by the Deutsche Forschungsgeeinschaft in the course of the project - Correct Model Transforations - see

2 1002 Orejas, Labers / Sybolic raph Transforation Keywords: Attributed graph transforation, sybolic graph transforation, lazy transforation 1. Introduction Attributed graphs and attributed graph transforation play a significant role in ost applications of graph transforation. In practice, an attributed graph transforation rule is like a noral rule, but soe nodes or edges are labelled by expressions over soe given variables. Then, defining a atch of a rule to a given object graph, whose attributes are soe concrete values, eans finding the values that ust 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 (e) in the object graph. That is, defining a atch of a rule eans solving a set of constraints. In any cases, these constraints are trivial. For instance, when a rule describes a transforation H, where the attributes in H are obtained by soe siple coputation fro the attributes in, e.g. when the expressions used as attributes in the left-hand side of the rules are just variables, and ore general expressions, defined over these variables, only occur in the right-hand side. However there are any other cases where the constraints to solve ay be not so trivial and, oreover, ay have several answers. For instance, when the transforation process includes soe 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 transforation these constraints ust be copletely solved when defining the atching of the given transforation rule. Then, finding a atch eans choosing one specific solution. This kind of early binding is well-known fro other areas of Coputer Science to be inadequate. One proble is that the solution chosen for the constraints associated to a given transforation step ay not be fully adequate, eaning that we ay need to backtrack this transforation step. The approach taken in areas like Constraint Logic Prograing [9], by which our approach is inspired, is to postpone solving the constraints as uch as possible, checking eanwhile their satisfiability. Then, not only ay we avoid soe useless backtracking, but we have other advantages. On the one hand, checking satisfiability ay be coputationally sipler than solving a set of constraints, eaning that it ay also be sipler to apply a transforation step. On the other hand, soe constraints which ay be difficult to solve at a given oent, ay becoe sipler, even trivial, because of the interaction with constraints defined by later steps. In [12], when studying the proble 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 liited expressive power for defining conditions on the attributes. To avoid these probles we presented a new foral approach which (we believe) is conceptually sipler and ore powerful than existing approaches. It is siple because, in our approach, attributed graphs are not defined as soe kind of cobination of a graph and an algebra, as in [10, 1, 7, 5], nor do we have to establish a difference between transforation rules and rule scheata, as in [16]. At the sae tie, our approach is expressively ore powerful, not only because we can define graph constraints with arbitrary conditions on the attributes as we aied, but also because we can define transforation rules that cannot be defined in other approaches, as shown in [14]. raphs in our approach are called sybolic graphs, because the attributes in the graph are represented sybolically by variables whose possible values are specified by a set of forulas. However, in this paper, we show that sybolic transforation rules, and the corresponding notion of

3 Orejas, Labers / Sybolic raph Transforation 1003 sybolic graph transforation, as defined in [14], are too restrictive for the transforation of arbitrary sybolic graphs. As a consequence, we introduce a ore general notion of sybolic graph transforation rules, which we have called narrowing rules, and characterize graph transforation by eans of these rules. In addition, we study the copatibility of sybolic graph transforation with the seantics of sybolic graphs 1, showing that this kind of transforation is copatible with the seantics of graphs, but not in a strong sense. Hence, we introduce a variant of sybolic graph transforation, which we have called, narrowing graph transforation, proving that it is strongly copatible with respect to the seantics of graphs. Finally, using the notion of narrowing graph transforation, we present a new approach to sybolic graph transforation that allows us to delay constraint solving when doing attributed graph transforation. In particular we show that this new approach is sound and coplete with respect to standard attributed graph transforation, and it is coplete and satisfies a property of extended soundness with respect to sybolic graph transforation. Moreover, a running exaple, where a graph transforation syste describes soe basic operations of a travel agency, shows the practical interest of the approach. The paper is organized as follows. In Section 2 we recapitulate soe notions that are used in the rest of the paper. In Section 3 we present the category and the seantics of sybolic graphs. Section 4 is dedicated to describe how (standard) sybolic graph transforation works. In Section 5 we present the new notions of narrowing rules and narrowing graph transforation studying its copatibility with respect to the seantics of sybolic graphs. Section 6 is dedicated to the new notion of lazy transforation and to prove its soundness and copleteness. Finally, in Section 6, we copare our approach with other related work and we draw soe conclusions. This paper is an extended version of [15], presented at ICT In particular, in addition to the detailed proofs of our results, we introduce the new notion of narrowing rules and narrowing transforation, presenting lazy transforation as a specific instance of narrowing transforation. 2. Preliinaries We assue that the reader has a basic knowledge on algebraic specification and on graph transforation. For instance, we advise to look at [6] for ore detail on algebraic specification or at [3] for ore detail on graph transforation Basic algebraic concepts and notation A signature Σ = (S, Ω) consists of a set of sorts S, and Ω is a faily of operation and predicate sybols typed over these sorts. A Σ-algebra A consists of an S-indexed faily of sets {A s } s S and a function op A (resp. a relation pr A ) for each operation op (resp. each predicate pr) in the signature. A Σ-hooorphis h : A A consists of an S-indexed faily of functions {h s : A s A s} s S couting with the operations and preserving the relations. iven a signature Σ, we denote by T Σ the ter algebra, consisting of all the possible Σ-(ground) ters. iven any Σ-algebra A there is a unique hooorphis h A : T Σ A. In particular, h A yields the value of each ter in A. Siilarly, T Σ (X) denotes the algebra of all Σ-ters with variables in X, and given a variable assignent σ : X A, this assignent extends to a unique hooorphis 1 This notion is defined in Section 5.

4 1004 Orejas, Labers / Sybolic raph Transforation σ # : T Σ (X) A yielding the value of each ter after the replaceent of each variable x by its value σ(x). In particular, when an assignent is defined over the ter algebra, i.e. σ : X T Σ, then σ # (t) denotes the ter obtained by substituting each variable x in t by the ter σ(x). However, for siplicity, even if it is an abuse of notation, we will write σ(t) instead of σ # (t) E-graphs and Attributed raphs 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 ay be decorated with labels fro a given set. The difference with labelled graphs, as coonly understood, is that in labelled graphs it is usually assued that each node or edge is labelled with a given nuber of labels, which is fixed a priori. In the case of E-graphs, each node or edge ay have any arbitrary (finite) nuber of labels, which is not fixed a priori. Actually, in the context of graph transforation, the application of a rule ay change the nuber of labels of a node or of an edge. Forally, 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 eans 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-raphs and orphiss) An E-graph over the set of labels X is a tuple = (V, X, E, E NL consisting of:, EEL V and X, the sets of graph nodes and of label nodes, respectively., s, s NL, sel, t, t NL E, E NL, and EEL, the sets of graph edges, node label edges, and edge label edges, respectively. and the source and target functions:, tel ) s : E V and t : E V s NL s EL : ENL : EEL V and t NL : ENL E and t EL : E EL X X An E-graph is linear if every label x is attached to at ost one node or one edge, i.e. there is at ost one (node or edge) label edge e such that x = t j (e), where j {NL, EL}. iven the E-graphs and, an E-graph orphis f : is a tuple, f V : V V, f X : X X, f E : E E, f E NL : E NL E NL, f E EL : E EL E EL such that f coutes with all the source and target functions. E-graphs and E-graph orphiss for the category E raphs. 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 seantics of a sybolic graph (cf. Definition 3.3):

5 Orejas, Labers / Sybolic raph Transforation 1005 Definition 2.2. (Label substitution) iven an E-graph = (V, X, E, E NL, EEL, s, s NL, sel, t, t NL, tel ), a set of labels X, and a function h : X X we define the graph resulting fro the substitution of X along h, h() = (V, X, E, E NL, s, s NL, s EL, t, t NL, t EL ), where:, E EL V = V, E = E, E NL t = t = E NL, E EL = E EL, s = s, s NL = s NL, s EL = s EL, and For every e E NL For every e E EL : t NL : t EL (e) = h(tnl (e)) (e) = h(tel (e)) Moreover, h induces the definition of the orphis h : h(), with h = id V, h, id E, id ENL, id EEL. Notice that if f : H is a orphis such that f V, f E, f ENL, and f EEL are bijections, then we can consider that f is induced by the label substitution f X (up to isoorphis). We have used the sae notation for the orphis 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 siplifies notation. each() eliinates all the labels in an E-graph which are not bound to a node or an edge and estr(, Y ) eliinates all the labels which are not in Y, assuing that Y includes all reachable labels (and, perhaps, soe unreachable ones). These constructions are used to define lazy graph transforation (cf. Definition 6.1). Definition 2.3. (eachable subgraph and restriction subgraph) iven an E-graph = (V, X, E, E NL, E EL, {s j, t j } j {,NL,EL} ), we define the reachable subgraph of, each() as the largest subgraph of where X each() consists of all labels x such that: There is a node label edge nl E NL such that x = t NL (nl), or there is an edge label edge el E EL such that x = t EL (el). Moreover, given a orphis f : we denote by each(f) : each() the restriction of f to the subgraph each(). Siilarly, we define the restriction of to a set of labels Y, estr(, Y ), where X each() Y as the largest subgraph of where X estr(,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 assued to be included in the graph. Definition 2.4. (Attributed graphs and orphiss) An attributed graph over Σ is a pair, D, where D is a given Σ-algebra, called the data algebra of the graph, and is an E-graph such that the set X of labels in consists of all the values in D, i.e. X = s S D s, where s is the set of sorts of the data algebra and denotes disjoint union. iven the attributed graphs over Σ A =, D and A =, D, an attributed graph orphis h : A A is a pair h graph, h alg, where h graph is an E-graph orphis, h graph : and h alg is a Σ-hooorphis, h alg : D D such that the values in D are apped consistently by h graph and h alg, i.e. for each sort s S the diagra below coutes:

6 1006 Orejas, Labers / Sybolic raph Transforation D s h alg D s X h graph X Attributed graphs and attributed graph orphiss for the category Attraphs. 3. Sybolic graphs In this section, we provide a sall introduction to sybolic graphs. In particular, in the first subsection, we introduce the category of this kind of graphs and describe soe results that can be found in [14]. Then, in the second subsection, we study the seantics of sybolic graphs presenting soe new technical results which are used in the rest of the paper The category of sybolic graphs If we consider that an attributed graph is like an E-graph whose labels are values over a given data algebra, a sybolic graph can be seen as the specification of a class of attributed graphs. In particular, a sybolic graph consists of an E-graph whose labels are variables, together with a set of forulas Φ that constrain the possible values of these variables. We consider that a sybolic graph denotes the class of all attributed graphs where the variables in the E-graph have been replaced by values that ake Φ true in the given data doain. For instance, the sybolic graph in Figure 1 specifies a class of attributed graphs, including distances in the edges, that satisfy the well-known triangle inequality. We ust 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 d 1, d 2 and d 3 to the corresponding graph edges. Actually, in the rest of the paper we will always oit depicting (node or edge) label edges, since we think that the resulting graphs are ore intuitive. d 3 d 1 d 2 with d 3! d 1 +d 2 Figure 1. A sybolic graph Definition 3.1. (Sybolic graphs and orphiss) A sybolic graph S over the data Σ-algebra D, with Σ = (S, Ω), is a pair S =, Φ, where is an E-graph over an S-sorted set of variables, used as labels, X = {X s } s S, i.e. X = s S X s, and Φ is a set of first-order Σ-forulas with free variables in X and including eleents in D as constants

7 Orejas, Labers / Sybolic raph Transforation 1007 iven sybolic graphs 1, Φ 1 and 2, Φ 2 over D, a sybolic graph orphis h : 1, Φ 1 2, Φ 2 is an E-graph orphis h : 1 2 such that D = Φ 2 h(φ 1 ), where h(φ 1 ) is the set of forulas obtained when replacing in Φ 1 every variable x 1 in the set of labels of 1 by h X (x 1 ). Sybolic graphs over D together with their orphiss for the category Sybraphs D. As said in the definition, we consider that Φ is a set of arbitrary first-order Σ-forulas. However, for practical purposes, we ay want to restrict the class of forulas that can occur in this set, since first-order forula satisfiability is an undecidable proble. In principle, the only condition that we need to ensure that the results presented in this paper hold is that the given class of forulas 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 sybolic graphs, as proved in [14]. eark 3.1. (Sybolic typed graphs) Even if along the paper, for siplicity, we have only considered untyped graphs, in our running exaple (see Exaple 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 orphiss S T, where S is a sybolic graph and T is the given type graph, which is a specific sybolic graph (i.e. the category of typed sybolic graphs over the type graph T is the slice category over T ). In particular, type graphs T ust be sybolic graphs T = ET, False, where ET is an E-graph over a set of labels X ET consisting of just one variable for each sort in the given data signature, and False is the set consisting just of the false forula. In this way, given a sybolic graph S =, Φ, any (typing) orphis t : ET for is also a (typing) orphis t : S T. In [14], we showed that sybolic graphs are an adhesive HL category taking as M-orphiss all injective graph orphiss where the forulas constraining the source and target graphs are equivalent (in ost cases they will just be the sae forula). Definition 3.2. (M-orphiss) An M-orphis h :, Φ, Φ is a onoorphis such that X = X, i.e. h X is a bijection, and D = h(φ) Φ We will not study in detail all the constructions and results that are needed to prove that Sybraphs D is an adhesive HL category (the interested reader is addressed to [14]). However, it is iportant to see how pushouts work in order to understand sybolic graph transforation: Proposition 3.1. [14] Diagra (1) below is a pushout if and only if diagra (2) is a pushout in E raphs and D = Φ 3 (g 1 (Φ 1 ) g 2 (Φ 2 )). 0, Φ 0 h 1 1, Φ 1 0 h 1 1 h 2 (1) g 1 h 2 (2) g 1 2, Φ 2 g 2 3, Φ 3 2 g 2 3 Therefore, as said above, we have: Theore 3.1. [14] Sybraphs D is adhesive HL.

8 1008 Orejas, Labers / Sybolic raph Transforation One ay wonder whether Sybraphs D is an adhesive category, in particular if it is adhesive HL when M-orphiss are just onoorphiss. The answer is no, as it can be seen fro the counterexaple below: Exaple 3.1. Let us consider the diagra below, where all the arrows are the identity graph orphis. Then, it is easy to see that the botto diagra is a pushout in Sybraphs D, but it is not a van Kapen square [11]., {true}, {true}, {false}, {false}, {true}, {false}, {false}, {false} In particular, we ay easily check that all the orphiss in the diagra are onoorphiss in Sybraphs D. Moreover, we can also check easily that the top and botto squares are pushouts and the back squares are pullbacks 2. However the front-left square is not a pullback. In particular, true false false. Moreover, we ay see that, in general, for diagras along onoorphiss, pushout copleents are not unique as the following counterexaple shows: Exaple 3.2. Let us consider the orphiss f :, {true}, {false} and g :, {false}, {false}, where both orphiss are identities as E-graph orphiss, then it is easy to see that the diagras below are pushouts:, {true} f, {false}, {true} f, {false} (1) g, {false}, {false}, {true}, {false} (2) g 3.2. Seantics of sybolic graphs As we said above, we ay consider that a sybolic graph S denotes a class of attributed graphs. In particular the class of all attributed graphs that can be obtained by replacing the variables in S by values 2 If the given forulas do not include variables, pullbacks in Sybraphs D are defined in ters of pullbacks in E raphs, where the associated forula of the resulting graph is the disjunction of the forulas of the graphs involved.

9 Orejas, Labers / Sybolic raph Transforation 1009 in the given data algebra, so that the associated condition becoes true in the algebra: Definition 3.3. (Seantics of sybolic graphs) The seantics of a sybolic graph, Φ over a data algebra D is a class of attributed graphs defined: Se(, Φ ) = { σ(), D σ : X D and D = σ(φ)} where σ() denotes the graph obtained according to Def We ay notice that the class of attributed graphs denoted by a sybolic graph ay be epty if the associated condition is unsatisfiable. Every attributed graph ay be seen as a sybolic graph by just replacing all its values by variables and by including an equation x v = v, for each value v in the data algebra, into the corresponding set of forulas, where x v is the variable that has replaced the value v. We call these kind of sybolic graphs grounded sybolic graphs. For instance, in Figure 2, on the right, we can see the sybolic 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 the by variables and showing the corresponding equation. This is done, for instance, in Exaple 3.3, where a grounded sybolic graph is displayed as an attributed graph d 3 d 2 12 d 1 with d 1 = 12, d 2 = 15, d 3 = 18 Figure 2. Attributed graph and grounded sybolic graph Definition 3.4. (rounded sybolic graphs) A sybolic graph, Φ over a data algebra D is grounded if X includes a variable, which we denote by x v, for each value v D, and 2. For every substitution σ : X D, such that D = σ(φ), we have σ(x v ) = v, for each variable x v X. Moreover, we define Sybraphs D as the full subcategory of Sybraphs D consisting of all grounded graphs.

10 1010 Orejas, Labers / Sybolic raph Transforation It should be obvious that the seantics of a grounded graph includes exactly one attributed graph, and that grounded graphs are closed up to isoorphis. Moreover, in [14] we can see that for every attributed graph A there is a unique grounded sybolic graph (up to isoorphis) S(A) such that Se(S(A)) consists of A. In particular, the E-graph associated to S(A) is obtained substituting every data value v in a set of labels by a variable x v, and the set of forulas in the sybolic graph consists of an equation x v = 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 sybolic graphs over D are equivalent, provided that the given data algebra is finitely generated. As a consequence, we ay identify the. For instance, we ay consider that the seantics of a sybolic graph is a class of grounded sybolic graphs. We ay notice that, if h : S S is a sybolic graph orphis and S is grounded then the condition D = Φ h(φ), that the sybolic orphis h ust satisfy, can be represented as a constraint satisfaction proble: Fact 3.1. iven a sybolic graph S =, Φ and a grounded sybolic graph S =, Φ, an E-graph orphis h : is a sybolic graph orphis if and only if D = h (Φ), where h : X D is the apping defined x X : h (x) = v if h(x) = x v. Proof: By definition, we know that h is a sybolic graph orphis if and only if D = Φ h(φ). This is equivalent to showing that, for every substitution σ, if D = σ(φ ) then D = σ(h(φ)). Now, Φ consists of all the equations x v = v, therefore the only substitution σ such that D = σ(φ ) is σ(x v ) = v. But for every variable x in X, h (x) = σ h(x). Therefore, this is equivalent to D = h (Φ). The above fact eans that defining a orphis fro S =, Φ into a grounded sybolic graph can be seen partially as a constraint satisfaction proble. In particular, the orphis ust provide values to the variables in Φ so that the conditions (constraints) in Φ are satisfied. For instance, in Exaple 4.2 we see an object graph that represents a custoer of a travel agency that wants to ake a hotel reservation for two days in Berlin and has a certain budget. Then, in that exaple, applying a transforation rule that is supposed to ipleent this reservation (i.e. defining the atch orphis) iplies 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 sybolic graph S and the grounded representation of any attributed graph A in the seantics of S there exists a sybolic orphis between S and S(A), which is induced by a label substitution. Fact 3.2. An attributed graph A =, D is in the seantics of S =, Φ if and only if there is a sybolic graph orphis h : S S(A) induced by a label substitution. Proof: If A Se(S) this eans that = σ( ) for a given variable substitution σ such that D = σ(φ). But this eans that we ay define h as the orphis associated to σ σ, where σ is the substitution that aps every value v D to the variable x v. In addition, D = σ(φ) iplies, by Fact 3.1, that h is a sybolic graph orphis. Conversely, if there is a sybolic orphis h : S S(A) induced by a given label substitution, we can define an assignent σ : X D with σ(x) = v if h(x) = x

11 Orejas, Labers / Sybolic raph Transforation 1011 and the equation x = v is in Φ S(A). Then, by Fact 3.1, we know that D = σ(φ) iplying that the attributed graph σ( ), D is in Se(S). The second fact states that, if A is in the seantics of S and the variables in S are a subset of the variables of S then any orphis f fro a sybolic graph S to S(A) can be factorized into two orphiss f : S S and h : S S(A), where h is the orphis defined in the previous fact. Fact 3.3. iven sybolic graphs S =, Φ and S =, Φ, such that is linear (cf. Definition 2.1), and given a orphis f : S S(A), where A Se(S ), there is an E-graph orphis f : S S such that f = h f, where h : S S(A) is the orphis defined in Fact 3.2. Proof: It is enough to define f as follows: For every graph node or edge e, f (e) = h 1 (f(e)). For every label x X, f (x) = x, if x is the target in of soe node label edge or of soe edge label edge e (i.e. x is attached to a node or an edge via e) and the target of f (e) in is x. For every label x X, f (x) = x,if x is not attached to any node or edge in and x is any label in such that f(x) = h(x ). It is routine to check that f is an E-orphis because, on the one hand, if h is the orphis induced by a label substitution, h is a bijection on graph nodes and edges, and on the other hand, is linear, which eans that the definition of f on the labels is correct. Moreover, by definition f = h f. Exaple 3.3. (Introduction to unning Exaple) In our running exaple we specify a travel agency in ters of typed sybolic graphs and sybolic graph transforation rules. In particular, the data signature of our exaple is: Sorts Opns int, bool, city, hotel, flight price : hotel int location : hotel city price : flight int departure : flight city destination : flight city and the data algebra, depicted by eans of two tables in Figure 3. The type graph, depicted in Figure 4, consists of a Custoer that can request a hotel reservation Hotelequest or a flight reservation Flightequest. These requests can be responded by a Flighteservation or a Hoteleservation, respectively. A Custoer has the labels nae and bud, describing the total budget that the custoer is willing to spend for his reservations. A Hotelequest has as labels the nuber of nights and loc, describing the city where the hotel is located. A Flightequest has as labels

12 c : Custoer 1012 Orejas, Labers / Sybolic raph Transforation hotel price city flight price departure destination h1 100 München h2 90 Berlin h3 85 Berlin h4 70 Köln h5 200 Berlin book flight book hotel «uses» «uses» book travel f1 175 Berlin Barcelona f2 115 Berlin Barcelona f3 150 Barcelona Berlin f4 85 Barcelona Berlin book transfer Figure 3. A data algebra «uses» Traveler Custoer -nae : string -bud : int int, bool, city, hote Hotelequest -loc : city -nights : int Flightequest -dep : city -dest : city price:hotel->int location: hotel -> price:flight->int departure: flight - destination: flight Hoteleservation -h : hotel Flighteservation -f : flight Figure 4. Type graph dep and dest, the departure and destination city of fr : the Flightequest flight, respectively. reserveflight The Hoteleservation fr : Flightequest and Flighteservation have only one label h and f for the dep hotel : city and flight that are reserved, respectively. dep : city dest : city dest : city Finally, a possible start graph describing the needs of a custoer called Alice is depicted in Figure 5. In particular, this graph is assued to be grounded c : Custoer 3 and shows a custoer with a budget c : Custoer of 450, requesting a hotel for two nights in Berlin, a flight nae fro: Barcelona string to Berlin and a flightnae fro: string Berlin to bud : int bud : int Barcelona. : Flighte f : flight 3 As explained above, for readability, it is depicted as an attributed graph dep = departure(f) && dest = destination(f) && bud = bud - price(f) && hr : Hotelequest loc : city nights : int reservehotel hr : Hotelequest loc : city nights : int : Hotelese h : hotel

13 0,-./1*8*,0/ 0,-./1*8*,0/.*8*.(/%& 9*8*?31/(;%" 0#;%*8*1/",0- '36*8*,0/ 9*8*?31/(;%" 0#;%*8*1/",0- 815$%$815&(/10*9,$--$2)!3$%$2)!$: *#'/5"*9,$;$0/<9(.,$--$2)!36%7 Orejas, Labers / Sybolic raph Transforation /#"/*-"#7.8 8*A(/%&=%>3%1/ &(9*8*9,/:*D*E%"&,0 0,-./1*8*,0/*D*F 8*<&,-./=%>3%1/ 6%7*8*9,/:*D*E#"9%&(0# 6%1/*8*9,/:*D*E%"&,0 8*?31/(;%" 0#;%*8*1/",0-*D*&,9% '36*8*,0/*D*HIJ 8*<&,-./=%>3%1/ 6%7*8*9,/:*D*E%"&,0 6%1/*8*9,/:*D*E#"9%&(0# Figure 5. Start graph 4. Sybolic graph transforation for attributed graph transforation In the previous section we have seen that the category of sybolic graphs is adhesive HL. Hence, following [3], we can define a first notion of sybolic graph transforation using spans of M-orphiss. This is equivalent to consider that the left and right-hand sides (and also the interface) of a rule are constrained by the sae set of forulas. This eans that we ay denote sybolic graph transforation rules as pairs L K, Φ, where L, K and are E-graphs over the sae set of labels X and Φ is a set of forulas 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 ay also ipose soe constraints on the atchings. Moreover, for technical reasons we require that L is linear (cf. Definition 2.1), eaning that no label is bound to two different eleents in L 4. Definition 4.1. (Sybolic graph transforation rule) A sybolic graph transforation rule is a pair L K, Φ, where L, K, are E-graphs over the sae set of labels X, L is linear, L K is a span of E-graph inclusions 5, and Φ is a set of forulas over X and over the values in the given data algebra D. Exaple 4.1. (Sybolic graph transforation rules) In Figure 6 we can see the rules that describe soe operations of the travel agency of our exaple. In particular, rule reserveflight connects a Flighteservation with a Flightequest for soe Custoer. Identical naes in rules specify that the corresponding nodes are preserved. Edges between nodes with identical naes are preserved as well. The rule includes a forula Φ, expressing that the departure city dep and destination city dest of the Flightequest 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 custoer is diinished 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 Hoteleservation with a Hotelequest for soe Custoer. It holds a forula Φ, expressing that the location loc of the Hotelequest should be identical to the location of the reserved hotel location(h). The total budget bud of the custoer is diinished by the price of the reserved hotel ultiplied with the 4 If needed we can transfor SL into an equivalent linear graph (with the sae seantics). We just need to change the repeated occurrences of the sae variable by new variables and add the corresponding equalities to the given set of forulas. 5 or, in general, onoorphiss

14 Hotelequest -loc : city -nights : int Custoer -nae : string -bud : int Flightequest -dep : city -dest : city 1014 Orejas, Labers / Sybolic raph Transforation int, bool, city, hotel, flight price:hotel->int location: hotel -> city price:flight->int departure: flight -> city destination: flight -> city flight -price : string -source -destination hotel -price : string -location Hoteleservation -h : hotel Flighteservation nuber of nights price(h)*nights -f : flight and as an extra constraint it is required again that the new budget bud is still greater than or equal to zero. Notice that the fact that these rules are spans of M-orphiss iplies that the variable f in the first rule ust be considered to be included in the left-hand side of the rule, even if it is not bound to any node or edge of the left-hand side graph. The sae happens with the variable h in the second rule, as well as with bud in both rules. city Custoer -nae : string -bud' : int fr : Flightequest dep : city dest : city reserveflight fr : Flightequest dep : city dest : city : Flighteservation f : flight c : Custoer nae : string bud : int c : Custoer nae : string bud' : int dep = departure(f) && dest = destination(f) && bud = bud - price(f) && bud >=0 hr : Hotelequest loc : city nights : int reservehotel hr : Hotelequest loc : city nights : int : Hoteleservation h : hotel c : Custoer nae : string bud : int c : Custoer nae : string bud' : int loc = location(h) && bud = bud (price(h) * nights) && bud >=0 start graph: : Hotelequest loc : city nights : int : Flightequest dep : city dest : city Figure 6. raph transforation rules : Flightequest As usual, : the Custoer application of a graph dep : transforation city rule L K, Φ to a given sybolic nae : string dest : city bud : int graph S can be defined by a double pushout in the category of sybolic graphs: Definition 4.2. (Sybolic graph transforation) iven sybolic graphs S and SH over a Σ-algebra D, a sybolic graph transforation rule, r = L K, Φ r over D, and a orphis : L, Φ r S, we say that SH is a direct transforation of S by r via, denoted S = r, SH, if SH can be obtained by the following double pushout in Sybraphs D : L, Φ r S K, Φ r SF, Φ r SH

15 Orejas, Labers / Sybolic raph Transforation 1015 Moreover, we write S = SH if the given rule and atch can be left iplicit, and we write = to denote the reflexive and transitive closure of =. The proposition below states soe properties that are iportant to understand sybolic graph transforation. The first property states that the set of labels and the condition of the given object graph reain invariant after sybolic transforation. The second property states that sybolic transforation preserves groundedness. The third property shows that the application of a sybolic transforation rule can be defined in ters of a transforation of E-graphs. Finally, the last condition states that if we can apply a transforation to a given graph S, then we can also apply (essentially) the sae transforation to a graph that extends S with soe additional variables and whose condition is ore restrictive. Proposition 4.1. iven a sybolic graph transforation rule r = L K, Φ over a given data algebra D: 1. If, Φ = r, H, Φ then X = X H and D = Φ Φ. 2. If, Φ = r, H, Φ and, Φ is grounded then H, Φ is also grounded. 3. If : L is an E-graph orphis, we have that, Φ = r, H, Φ if and only if the diagra below is a double pushout in E raphs and D = Φ (Φ). L K F H 4. If, Φ = r, H, Φ, X X, and D = Φ Φ then X, Φ = r,i H X, Φ, where X denotes the graph obtained by adding to all the labels that are in X but not in X and i is the inclusion i : X. Proof: 1. According to the definition of sybolic graph transforation rules, X L = X K = X iplying that the orphiss in the span L, Φ K, Φ, Φ are M-orphiss. If, Φ = r, H, Φ, this eans that the diagra below is a double pushout in Sybraphs D : L, Φ K, Φ, Φ, Φ F, Ψ H, Φ But we know [14] that pushouts in Sybraphs D preserve M-orphiss (see [14]), hence the orphiss in the span, Φ F, Ψ H, Φ are also M-orphiss. This eans that X = X F = X H and D = Φ Ψ Φ.

16 1016 Orejas, Labers / Sybolic raph Transforation 2. A direct consequence of the previous ite. The reason is that, if Φ defines the values of all the variables in X, then Φ also defines the values of all the variables in X H = X. 3. If, Φ = r, H, Φ this eans that is a sybolic graph orphis and the diagra below is a double pushout in Sybraphs D : L, Φ K, Φ, Φ (1), Φ F, Φ (2) H, Φ Now, according to Proposition 3.1, this eans that the diagra below is a double pushout in E raphs. Moreover, since is a sybolic graph orphis then D = Φ (Φ). L K (3) F (4) H Conversely, let us suppose that diagras (3) and (4) are pushouts in E raphs. We know that if the orphiss in the span L K are the identity on variables then so are (up to isoorphis) the orphiss on the span F H. This eans that the orphiss, and coincide on the variables and, as a consequence, if D = Φ (Φ), then D = Φ (Φ) and D = Φ (Φ). Hence all the orphiss in (1) + (2) are sybolic graphs orphiss. Therefore, to prove that (1) and (2) are pushouts, according to Prop. 3.1, we have to prove that D = Φ Φ (Φ) and D = Φ Φ (Φ), but this is obvious if D = Φ (Φ) and D = Φ (Φ). 4. If, Φ = r, H, Φ this eans that D = Φ (Φ) and H can be obtained by the double pushout below: L K F H We know that i satisfies the gluing conditions with respect to r, since i just adds soe variables that are not connected to any node or edge. Therefore, we can build the following double pushout diagra: L K i X F X H X But, if D = Φ Φ and i( (Φ)) = (Φ), since i is the equality on all labels in X, then D = Φ i( (Φ)) iplying X, Φ = r,i H X, Φ.

17 Orejas, Labers / Sybolic raph Transforation 1017 The fact that the forula constraining the variables of the E-graph H after transforation is the sae forula for the graph before transforation does not ean that the attributes in the graph do not change after transforation. The reason is that the transforation ay change the bindings of variables to nodes and edges, as the exaple below shows. Moreover, according to Fact 3.1, if we apply the rule L K, Φ to a grounded sybolic graph S =, Φ, the condition D = Φ (Φ) can be seen as the constraint satisfaction proble where we find the values to assign to the variables in Φ, so that this set of forulas is satisfied in D. Exaple 4.2. In Figure 7 we can see how the rule reservehotel presented in Exaple 4.1 is applied to a given start graph as presented in Exaple 3.3, describing a custoer that would like to reserve a hotel for 2 days in Berlin and, in addition, would also like to reserve a flight fro Barcelona to Berlin and return. : Hotelequest loc : city = Berlin nights : int = 2 : Custoer nae : string = Alice bud : int = 450 : Flightequest dep : city = Barcelona dest : city = Berlin : Flightequest dep : city = Berlin dest : city = Barcelona reservehotel : Hoteleservation h : hotel = h5 : Hotelequest loc : city = Berlin nights : int = 2 : Custoer nae : string = Alice bud : int = 50 : Flightequest dep : city = Barcelona dest : city = Berlin : Flightequest dep : city = Berlin dest : city = Barcelona reserveflight : Hoteleservation h : hotel : Hotelequest loc : city = Berlin nights : int = 2 : Custoer Figure 7. : Flightequest dep : city = Barcelona dest : city = Berlin Berlin = location(h) && bud = 450 (price(h) * 2) && bud >=0 : Flightequest fr1 : Flighteservation : Hotelequest : Flightequest also fr1 : Flighteservation reserveflight be included in all the graphs loc : city = inberlin the dep rule. : city Then, = Barcelona when defining the atch orphis we ust bind nights : int = 2 dest : city = Berlin f : flight that variable to soe hotel hi in D : Hoteleservation hotel such that the forula h : hotel : Flightequest fr2 : Flighteservation : Custoer Berlin = location(hi) dep : city = Berlin f : flight nae &&: string bud =450-(price(hi)*2) = Alice && bud >=0 dest : city = Barcelona bud : int is satisfied. Here, Berlin = the location(h) chosen&& hotel, Barcelona h5, satisfies = departure(fr1.f) the above && Berlin forula = destination(fr1.f) for h=h5 and && Berlin bud =50. = This kind of departure(fr2.f) && Barcelona = destination(fr2.f) && bud = 450 (price(h) * 2) - price(fr1.f) - early binding ay not be very efficient. For instance, after the transforation defined above no flight price(fr2.f) && bud >=0 reservation is possible, because the budget of the user would not be sufficient anyore for reserving any flight connecting Barcelona and Berlin. Hence, we would need to backtrack this transforation and to hotel price location try a different hotel. This situation also int, bool, city, hotel, flight h1 occurs 100 when Münchenworking with attributed graphs and attributed graph transforation rules as in [3]. h2 90 Berlin h3 85 Berlin price:hotel->int As shown location: in hotel [14], -> sybolic city graph4transforation 70 Köln is ore powerful than attributed graph transforation as presented price:flight->int in [3]. In particular, any attributed graph transforation rule r can be represented by flight price departure destination a sybolic departure: graph transforation flight -> city rule S(r) such that an attributed graph A can be transfored by r f1 175 Berlin Barcelona destination: flight -> city f2 f3 f Sybolic transforation nae : string = Alice dep : city = Berlin In this transforation, a hotel in Berlin is chosen bud : int dest : city non-deterinistically. = Barcelona This choice is ade when defining the atching 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 (price(h) if it* is 2) not - price(fr1.f) explicitly && bud depicted, >=0 the left-hand side of eservehotel is supposed to include the variable h, because that variable is explicitly in the right-hand side of the rule and hence it should Berlin Barcelona Barcelona f : flight Barcelona Berlin Berlin Possible solutions in the for (h,fr1.f,fr2.f,bud):

18 1018 Orejas, Labers / Sybolic raph Transforation into a graph AH if and only if the grounded graph associated to A, S(A), can be transfored by S(r) into S(AH). This representation is quite obvious. If ar = AL AK A is an attributed graph transforation rule, usually this eans that AL, A and AK are graphs over the ter algebra, i.e. they include ters (with variables) as labels. This eans that we can, straightforwardly, represent r as a sybolic graph transforation rule S(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 nae conditions in the grounded Custoergraphs S(AL), S(A), and S(AK)) would be equations X t = t, string Custoer where t is a ter occurring as an attribute in the rule. In this context, as said above, it is proven that, given -nae : string attributed graphs A, AH, A = r, AH if and only if S(A) = bud -bud : int S(r),S() S(AH). However, the converse is not true. In [14], we also proved that there are sybolic int rules that cannot be siulated by attributed graph transforation rules. Hotelequest Flightequest Hotelequest -loc : city Flightequest -dep : city 5. -nights Narrowing : int rules and -dest narrowing : city graph transforation city h As we have seen above, sybolic graph transforation can be considered powerful enough to fspecify transforations of attributed graphs as studied in [3]. However, the kind of rules that we have studied in Hoteleservation Section Hoteleservation 4 are not powerful enough Flighteservation to specify transforations of arbitrary sybolic graphs that ay be considered -h : hotel reasonable. For instance, -f : flight given the start graph in Figure 8, we would be unable to apply the rule eservehotel to. The proble is that we cannot atch the variable h, which (as said above) is iplicitly in the left hand side of the rule, to any variable in. We could think that a possible solution is to assue that all sybolic graphs iplicitly include a denuerable set of variables of each sort, so that h can be atched to soe variable in of sort hotel 6. However, this is not enough. If f is supposed to be the atch orphis then the condition in (i.e. true) ust iply f(φ), where Φ is the condition of the rule. However, in general, this is obviously false, since true only iplies tautologies. loc hotel nights dep flight dest Flighteservation Custoer -nae : string -bud : int Hotelequest -loc : city -nights : int Figure 8. Start graph The proble is that a rule consisting of a span of M-orphiss can transfor 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 inforation to the graph, where this new inforation is represented by new variables and new conditions over the variables. To overcoe this proble, we consider a ore general kind of rules, which we call narrowing transforation rules, because, as we ay see below, when applied to a sybolic graph, it restricts or narrows their associated conditions by adding the conditions of the rule. These rules consist of a span of onoorphiss, where only the left orphis is assued to be an M-orphis. 6 We ay note that, in Exaple 4.2, the start graph does iplicitly include a denuerable set of variables, even if they are not depicted in Figure 5. The reason is that, forally, that graph is assued to be grounded, which eans that it includes a variable per each existing data value.

19 Orejas, Labers / Sybolic raph Transforation 1019 Definition 5.1. (Narrowing graph transforation rule) A narrowing graph transforation rule over a Σ-algebra D is a triple Φ L, L K, Φ, where L, K are E-graphs over the sae set of labels, L is linear, L K is a span of E-graph inclusions 7, which eans that X L = X K X, Φ L and Φ are sets of Σ-forulas over X L and X, respectively, and over the values in D, and finally D = Φ Φ L. Note that the fact that the orphis K, Φ L L, Φ L is an M-orphis guarantees that the pushout copleent in a rule application is unique, if it exists (see Exaple 3.2). This eans that we ay define sybolic graph transforation, using narrowing rules, by double pushouts, in the standard way: Definition 5.2. (Sybolic graph transforation with narrowing rules) iven sybolic graphs S and SH over a Σ-algebra D, a narrowing rule, r = Φ L, L K, Φ over D, and a orphis : L, Φ L S, we say that SH is a direct transforation of S by r via, denoted S = r, SH, if SH can be obtained by the double pushout in Sybraphs D : L, Φ L K, Φ L, Φ S (1) SF and Se(SH) is not epty. As usual, we write S = SH if the given rule and atch can be left iplicit, and we write = to denote the reflexive and transitive closure of =. We ay notice that, in principle, the conditions Φ SH of the sybolic graph SH obtained by the above double pushout if, Φ = r, H, Φ SH, could be unsatisfiable in D, which eans that the seantics of H, Φ (Φ ) would be epty. Obviously, it akes little sense to ake a transforation whose result is inconsistent. For this reason, this is forbidden in this definition. We ay also notice that, if S = r, SH, S is a grounded graph and r is a narrowing rule then SH ay be not grounded, because the conditions added by the application of r ay be satisfied by several non-isoorphic graphs. As in the case of transforations by (standard) sybolic rules, sybolic graph transforations can be defined in ters of E-graph transforations. As in Proposition 4.1, this is just a consequence of how pushouts in Sybraphs D are defined. Fact 5.1. iven a narrowing transforation rule r = Φ L, L K, Φ over D and an E-graph orphis : L, we have that, Φ = r, H, Φ (Φ ) if and only if the diagra below is a double pushout in E raphs and D = Φ (Φ L ). L K (2) SH (1) F (2) H eark 5.1. Obviously, narrowing rules are a generalization of sybolic graph transforation rules, in the sense that every sybolic rule r = L K, Φ r can be seen as the narrowing rule 7 or, in general, onoorphiss