Mapping metamodels and models to database tables





~

z_new| ∃~z∈M inus, ∀k∈Z⁺_n : ~z_new[A_j] =y∧^

j6=k

~z_new[A_k] =~z[A_k]





 ,

whereZ⁺_n denotes the set of positive integers up ton(i.e.,1≤k≤n).

Definition 47 Theinsert operation

INSERT INTOT(A1, . . . , Ak)VALUES(y1, . . . , yk)

adds an n-tuple~yto tableT, if~yis not yet contained. The tuple~yhas valuey_iin columnA_i, respectively, and it contains undefined values in all other columns. In other words,T⁰ =T∪ {~y}, where~y[Ai] =yi, if1≤i≤k, and~y[C] =ε, ifC /∈ {A₁, . . . , A_k}.

Definition 48 Given a sequence of database operationsT A, atransaction is executed on a represen-tationMresulting in an other representationM⁰ (denoted by M =^{T A}⇒ M⁰), if either all operations ofT Aor none of them are executed.

6.4 Graph transformation in relational databases

We formally present how a graph transformation engine (following the single pushout [118] approach with injective matchings) can be implemented using a relational database. First, we describe how an ap-propriate database schema can be created based on the metamodel, and how the database representation of the model can be generated (Sec. 6.4.1). Afterwards, the pattern matching phase of rule application is implemented using database queries (Sections 6.4.2 and 6.4.3), finally data manipulation is handled (in Sec. 6.4.4).

6.4.1 Mapping metamodels and models to database tables

Mapping of metamodels to database tables. Instance models representing the system under design are stored in database tables. We use the standard bi-directional mapping (for more details see [49, 110]) to generate the schema of the database with BCNF property [27] from the metamodel.

• Let us first introduce a set calleddatabase identifier universe(denoted byU^d), which denotes the set of all identifiers that might be stored in the database.

• Each class Cwithkoutgoing many-to-one associations (C →^A1₁ C1, . . . , C ^A→^k₁ Ck) is mapped to a table withk+ 1columnsC^d(id, A^d₁, . . . , A^d_k).

– Columnidwill store the identifiers of objects of the specific class.

– ColumnA^d_i will contain the identifiers of target objects of such outgoing many-to-one links that have association C →^Ai₁ Ci as their direct type. If no such outgoing link exists in the model, the undefined valueεis used in columnA^d_i.

Additionally, we should define foreign keys∀i∈[1..k] : C^d.A^d_i ^{F K}→ C_i^d.idto respect the graph structure in the database. Formally,C^d⊆U^d× C₁^d∪ε

×. . .× C_k^d∪ε .

6.4. GRAPH TRANSFORMATION IN RELATIONAL DATABASES 87

• We assign a tableA^d(src, trg)for each many-to-many associationCs A

→_∗ Ctconnecting classes C_s and C_t in the metamodel. Columns src and trg contain identifiers of source and target objects, respectively. Foreign keysA^d.src^{F K}→ C_s^d.idandA^d.trg^{F K}→ C_t^d.idshould additionally be defined to respect the graph structure (preserve the source and the target of edges) in the database. In a more formal way,A^d⊆C_s^d×C_t^d.

• If a classCis inherited from a superclassD, then tableC^dshould be extended by a foreign key constraintC^d.id^{F K}→ D^d.id.

We introduced the superscriptdto uniformly denote database representations of all kinds of graph transformation related entities. For instance,C^d,r_LHS^d , andc^dmark the entities that represent a class C, a rule graph rLHS, and an object c in the database, respectively. This notation is always used as a bi-directional mapping meaning that, e.g., C^d unambiguously identifies the database table that was assigned to classC, and vice versa.

Mapping of instance models into rows. Now we define a bijective mapping, which assigns an iden-tifier to each object of the instance model. The image of the mappingc^dwill be used as a primary key that identifies objectcin the database.

In order to appropriately represent an object in the database, its key has to be contained by all tables that are assigned to an ancestor of the object’s type. Since inheritance relation in the metamodel (i.e., the type hierarchy) poses restriction (in the form of foreign key constraints) on exactly the same set of tables, additional care has to be taken when inserting (or deleting) even a single key (identifier). The order that handles insertion correctly is being defined now.

Definition 49 Given a metamodelM Mwith inheritance relations that are acyclic, atopological order of a typet(denoted byT opologicalOrder(t)) is such a sequence of the ancestors oftin which a class Dcannot appear before an ancestorCin the order, ifC ^^∗ D.

A natural consequence of the definition is that typetis the last element in its topological order.

Definition 50 Given a metamodelM Mwith inheritance relations that are acyclic, aninverse topolog-ical order of a typet(denoted byInverseT opologicalOrder(t)) is a topological order ofttraversed in the opposite order.

A natural consequence of the definition is that typetis the first element in its inverse topological order.

After fixing a certain topological and inverse topological order of a type to be used in the sequel, Algorithm 6.1 derives the database representation of the initial model as follows.

• We suppose that all the tables are initially empty.

• A new identifierc^dis generated for each objectcof the instance modelM. Then ancestors of the typet(c)of object care determined and furthermore they are ordered topologically according to the inheritance relation. The ordering is done in a top-down manner, meaning that the “most general” class is enumerated first. (The role of topological ordering is to avoid the violation of foreign key constraints that have already been imposed on database tables.) The final step is to insert the new identifier to all the tables that have been assigned to the enumerated ancestor classes.

88 CHAPTER 6. GRAPH TRANSFORMATION IN RELATIONAL DATABASES

Algorithm 6.1From instance model to its database representation

1: for allc∈V_M {For all objects in modelM}do

2: c^d:=GenerateN ewIdentif ier()

3: for allC∈T opologicalOrder(t(c))do

4: INSERT INTOC^d(id)VALUES(c^d){Inserts the new identifier to all ancestor tables}

5: end for

6: end for

7: for alla→^e₁ b∈E_M {For all many-to-one links in modelM}do

8: UPDATEsrc(t(e))^dSETt(e)^d =b^dWHEREid =a^d{Updates the value in columnt(e)^dtob^din the row with identifiera^d}

9: end for

10: for alla→^e_∗ b∈E_M {For all many-to-many links in modelM}do

11: INSERT INTOt(e)^d(src, trg)VALUES(a^d, b^d){Inserts identifiers of end pointsaandbinto the table that corresponds to many-to-many associationt(e)}

12: end for

• For each many-to-one linka→^e ₁ bof the instance model, the row in the tablesrc(t(e))^d, which represents the source objecta, is updated by replacing the value in columnt(e)^dby the identifier b^dof the target objectb.

• For each many-to-many link a →^e _∗ b of the instance model, the identifiers of the source and target nodes (a^d andb^d) are inserted to the tablet(e)^dthat has been assigned to the edge type (association)t(e)of linke.

We introduce a new term that formalizes the consistent database representation of an instance model.

Definition 51 Let a metamodelM M, and a database schemaS_DB be given together with the bidirec-tional mappingdfromM Mto the tables ofS_DB.

A modelMand a database representationMare consistent(M ∼=M), if

• each object of the instance model is represented in the database by one row in all the tables that have been assigned to ancestors of the node type. Moreover, these rows must contain the identifier of the object in their identifier column id. Formally, ∀C ∈ VM M,∀c ∈ VM :

C ^^∗ t(c) ⇐⇒ ∃~c∈C^d : ~c[id] =c^d

,

• each many-to-one link of the instance model is represented in the database by exactly one row in the table that corresponds to the source class of the type of the edge. This single row must contain identifiers of source objects in the identifier column idand target objects in the column corresponding to the direct type of the edge. Formally, a →^e₁ b ∈ E_M ⇐⇒

∃~a∈src(t(e))^d : ~a[id] =a^d∧~a[t(e)^d] =b^d , and

• the identifiers of source and target nodes of each many-to-many link (edge) of the instance model can be found exactly in the table that corresponds to the type of the edge. Formally,a→^e_∗ b ∈ EM ⇐⇒ a^d, b^d

∈t(e)^d.

Finally, we formulate a theorem, which states that the database representation that has been created by the above-mentioned initialization algorithm is consistent with the initial instance model.

6.4. GRAPH TRANSFORMATION IN RELATIONAL DATABASES 89

Theorem 1 The initial instance model M and its database representation M are consistent (see Def. 51). Formally,M ∼=M.

PROOF Proofs of all theorems can be found in Appendix A.