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3.5.2 CDL DATA MODEL TYPES Specifications and Implementations. The CDL language only provides interface specification of concepts and modules. The choice of an actual programming language is independent of CDL itself, but a binding to Visual Basic was implemented.

Object Types. The CDL system is strongly typed. The two foundational metatypes are collections and structures (tuples). All collections are ordered multisets with items of the same type (values, objects or collections) plus any additional properties — often indices or user-defined aggregate data, while ODMG has multiple collection types without additional properties. Structure types Semantic Web as an Object-oriented Database 19 may be either simple literals, or objects with an OID. For simplicity and with regard to common usage, at most one collection type can be associated with each structure type.

All objects of the class below have an OID, so their NextTo attribute is a collection that contains references to other houses. The HasRooms attribute may represent an aggregated

collection of embedded objects without an OID, depending on its definition:

–  –  –

Subtyping and Inheritance. CDL supports full multiple inheritance of both value and object types. In subtypes, properties may be added, modified and even removed. In place of interfaces, CDL uses abstract classes without an implementation that can be used to specify contracts for other classes (but collection types of abstract classes are not abstract to enable collections of subclass instances).

The concept of roles allows objects to dynamically inherit additional types at runtime. For this purpose and to enable stronger type checking, subtypes are defined to be either supplementary or alternative.

For the Person class, subtypes Man and Woman are alternative — an object of one of these types may not gain the other type. On the contrary, subtypes like Student, Parent or Employee are supplementary because an object of type Person may gain or lose one or more of these types during its lifetime in the database.

Literals. A literal has no OID and has to be embedded in an object. All collections and simple types (like numbers) are literals, while structures may be defined as either literals or objects.

Extents and keys. For each object type, the designer may order the system to keep an extent. In that case, the associated collection type is used for storing the extent. An extent also contains all instances of subtype objects. Every persistent object in the system needs to in at least one extent.

Keys. Every property can become a key (MatchCode). The system makes sure that all values of the key within its extents are unique. Another modifier may specify that the property is required (Mandat).

Metadata. Information about types is contained in the Database object, which is stored along with all other data. All extents are also embedded in this object, along with other properties that describe the database.

The Car concept has a corresponding Cars collection with an AveragePrice attribute and an

index of car types. These types are also used for object labels:

–  –  –


Attributes. All attributes are named and strongly typed. They can be implemented algorithmically or as data members, and this can change in subtypes (ODMG does not allow algorithm implementations for properties of literals). An attribute can be a structure, a collection or an object (which creates a relationship), and it may be parameterized by a named typed parameter.

Relationships are attributes whose type is an object (1:1) or a collection of objects (1:N).

Additional information can include the name of an inverse relationship from the target (in ODMG, only these two-way links are called “relationships”); relationships can also be subordinate, which means that all targets are deleted upon deletion of the master object.

Constraints. The range modifier can limit the value of a property. Range can be given by an enumeration of legal values, a numeric interval, and for collections (such as in 1:N relationships), by the set of objects contained in another collection.

Operations include procedures and functions. They are similar to their ODMG counterparts.

In this concept, it is likely that TitledName will be a computed attribute without actual physical storage. HighestTitle must be one of employee’s titles, AverageWage is different for each year

and SendMessage is a function (probably with side-effects):

–  –  –


Object Identifiers are unique within the database system, immutable and system-assigned. Since OIDs are not human-readable, each structure definition can appoint one of its properties to serve as a label for its instances in the graphical browser.

Object Names can be defined for any structure type, even for values. This is done within the type definition; its extent must always contain these named objects. An empty structure with named values can represent an enumeration type.

In an enumeration type, only the names of instances are needed:

–  –  –

3.5.3 EXAMPLE OF FORMAL SEMANTICS CDL has been partially formalized in [Güttner03] recently and its model closely resembles the O2 one, although it has certain categorical extensions. For this reason, the following example is only a brief

excerpt from the model (and obviously builds on several previous definitions):

A database schema is a tuple (S, L, D, OID, A) satisfying these conditions:

D is the set of elementary domains, OID are OID sets, A are attribute names, L is the set of type labels, S(D, OID, A) is a function that assigns types.

∀t∈T: kind-of(t)= elem ⇒ t∈cod(S).

All elementary domains are part of the schema.

∀o∈OID ∃!t∈cod(S): kind-of(t)= ref ∧ t = (o,x).

Fr every set of OID references, there is a corresponding structured object type.

∀t∈cod(S): kind-of(t)= data ⇒ is-def-in(t,S∪OID).

No data type is defined using some unknown type.

∀t∈cod(S): kind-of(t)∈{ref,col} ∧ t=(u,v) ⇒ is-def-in(v,S∪OID).

No object or collection type includes an unknown type in its definition.

∀t∈cod(S): kind-of(t)=col ⇒ ∃!u∈cod(S): t=(Nu,v).

For every collection type, there is a basic type for objects that it contains.

∀t∈S: ¬(t is-part-of t).

The is-part-of function helps indicate infinite levels of embedded data within objects.

A full RDF-compatible formal description of an object model closely resembling CDL is in chapter 6.


CDL was implemented by Vema co. in its G2 object database several years ago.

The implementation was deployed in a number of commercial applications. It was based on a high-level interpreted language (Microsoft Visual Basic), which enabled features such as dynamic inheritance and algorithm storage in the database. Another advantage was natural support for complex collection types, while some of the disadvantages involved performance issues. After several years, the G2 database development was discontinued.

3.5.5 SOURCES [HM00] contains an introduction to CDL and [Güttner03] formalizes the core features of the language. The company that implemented G2 can be found at http://www.vema.cz/.



Category theory (CT) is a very general formalism with big potential for modeling complex systems in a declarative way. It is also radically different from the usual set- and logic-based approaches. Moreover, recent advances in graphical modeling (such as UML [OMG03]) are likely to find formal expression in CT’s fully graphical representation. Several object-oriented data models based on category theory were proposed and attempts at their implementation exist.

Semantic Web as an Object-oriented Database 22


A category C is defined as a collection of objects obj(C), arrows arr(C) and a composition function comp; arr(C) × arr(C) → arr(C). Every arrow f∈arr(C) has a source src(f)∈obj(C) and a target tgt(f) ∈ obj(C) (arrows are

written as f: src(f) →tgt(f)). Four additional conditions must hold:

Composition of arrows:

∀ f,g∈arr(C): tgt(f)=src(h) ⇒ ∃ h∈arr(C): src(h)=src(f) ∧ tgt(h)=tgt(g) ∧ h=comp(f,g)

Associativity of composition:

∀ f,g,h∈arr(C): comp(f,comp(g,h)) = comp(comp(f,g),h) if one of the sides is defined

Existence of identity arrow:

∀ x∈obj(C) ∃ idx∈arr(C): src(idx)=tgt(idx)=x

Function of identity arrow:

∀ f∈arr(C): comp(idsrc(f),f)=comp(f,idtgt(f))=f Special types of arrows and objects are defined depending on the structure of a category. These notions are universal and can be applied to any category, no matter what the arrows and objects represent. For an example of categorical reasoning, see the subsection about formal semantics.

The manifests of Diskin and Goguen ([DC94] and [Goguen91]) indicate several advantages for using

category theory for formal specification of object-oriented data models:

Category Theory (CT) offers a graphical, yet formal language for modeling objects. Generic reasoning about any one object simplifies abstraction and proofs.

CT provides uniform constructs that help discover interesting generalizations and see complex properties that would otherwise go unnoticed.

CT uses a declarative approach that specifies objects only in terms of their relationships with other objects. Unlike in set or logic theories, the internal structure of objects does not need to be given.

3.6.2 LDM DATA MODEL In this section, categorical LDM (Limit Data Model) of [Kolenčík98] is briefly compared to the ODMG model. Only a limited part of a full object model has been formalized, since the principal focus of LDM was to give a rigorous categorical background to several basic modeling concepts.

Types. LDM is strongly typed — all of its objects have at least one type. Some types can be referenced by relationships (classes) while others need to be embedded in other objects (literals).

All objects behave like tuples, because they are aggregated from simpler types and references.

Subtyping and Inheritance. LDM models true multiple inheritance and even dynamic inheritance using roles. In this context, it defines the semantics of attribute access, virtual classes, polymorphism, and method dispatching.

Attributes and Relationships. The semantics of attributes are identical to the ODMG standard.

Relationships are always unidirectional, and can be either 1:1 or 1:N; in the latter case, a set of OIDs is modeled by power objects. LDM also ensures referential integrity.

Operations are typed and their semantics is expressed by means of category theory using exponential objects. Method side effects are not part of the model.

–  –  –

Object Identifiers are strictly logical. They are represented by elements in the category of sets — each element is unique, and it can be connected to other elements by appropriate mappings.

Object Lifetimes. All objects in the model are considered persistent.

Queries. The LDM model formalizes a large part of OQL query language of ODMG–93, focusing on the select–from–where clause.


Some approaches to formalizing the persistent object-oriented model using CT are given below:

Product model of Nelson and Rossiter [NR95] focuses on queries, closure and views (using subcategories and natural transformations), but it also formalizes keys, relationships and aggregation. Each class is modeled as a separate category with methods and dependencies represented by arrows whose source and target represents persistent and in-memory variables.

Keys correspond to initial objects, relationships to pullback categories, generalization to a coproduct.

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