A UNIFYING FORMAL ONTOLOGY MODEL

A Simple Formal Model for Unifying the Presentation of Ontologies in Semantic

Web Research

Stephan Grimm

FZI – Research Center for Information Technology at the University of Karlsruhe

Haid-Und-Neu-Str, 10-14, 76131 Karlsruhe, Germany

Keywords:

Ontologies, Formal Model, Semantic Web.

Abstract: We propose a simple formal ontology model for the uniform presentation of ontologies across different areas

of SemanticWeb (SW) research. On the one hand, the models simplicity allows for abstracting from technical

details in a selective way, while on the other hand it captures the essential characteristics of an ontology com-

mon to most ontology languages and formalisms. To demonstrate the compatibility of our model to existing

SW research, we provide mappings to several languages and formalisations ranging from expressive language

standards to light-weight semantical models. Moreover, we specify an extension of the UML metamodel for

the graphical visualisation of ontologies, and we sketch a software architecture for the programmatic access

to ontologies in terms of an API.

1 INTRODUCTION

Ontologies play a key role in the Semantic Web (SW)

as conceptual yet computational models for the ex-

plicit representation of domain knowledge in infor-

mation systems. One particular aspect of ontologies

as computational knowledge representation artifacts

is their formality according to a well-known deﬁni-

tion in (Gruber, 1993), which allows for their auto-

mated processing by machines in a meaningful and

clearly deﬁned way based on the principles of for-

mal semantics. In various areas that inﬂuence the

SW, such as knowledge representation and reasoning,

ontology alignment or Web 2.0 and light-weight se-

mantics, however, the formal aspects of ontologies

are perceived in different ways, emphasising differ-

ent characteristics. In consequence, formal models

used to describe ontologies in these areas deviatefrom

each other and each sub-community has its own for-

mal deﬁnition of an ontology.

To mediate between the various areas in the SW

community, striving for synergy in different strands

of ontology research, it would be beneﬁcial to have a

common basis for a formal model of an ontology that

provides a uniﬁed view on the subject across different

perspectives of processing techniques and use cases.

Such a common formal model should, on the one

hand, be powerful and expressive enough to capture

all the speciﬁc characteristics of an ontology across

the various ﬁelds. On the other hand, such a model

should not be overly formal in cases where certain as-

pects are to be abstracted from, providing a simple

view that is easily and intuitively understood by re-

searchers from all areas.

Besides the unifying framework for the technical

formalities of an ontology that such a model provides,

there are aspects of usage towards graphical visuali-

sation of and programmatic access to ontologies.

In this paper, we propose a unifying formal model

for the presentation and investigation of ontologies

across SW research strands. We demonstrate its com-

patibility to current SW research by providing map-

pings to existing ontology languages and formalisa-

tions in different SW areas. Moreover, we sketch a

UML metamodel extension for graphical visualisa-

tion of and a software API for programmatic access to

ontologies, both in accordancewith our formal model.

2 ONTOLOGY ESSENTIALS

According to the well-established deﬁnition in (Gru-

ber, 1993), an ontology is a “

formal explicit speciﬁ-

cation of a shared conceptualisation of a domain of

interest

”. Hence, it combines various aspects, such

327

Grimm S. (2009).

A UNIFYING FORMAL ONTOLOGY MODEL - A Simple Formal Model for Unifying the Presentation of Ontologies in Semantic Web Research.

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development, pages 327-335

DOI: 10.5220/0002331003270335

 SciTePress

hasPart

Engine

Car Ship

Vehicle

Maserati

differentFrom

isA

hasPart

250

horsePower

Figure 1: An example ontology as semantic network.

as formality and explicitness of represented knowl-

edge for machine-processing, consensus for commu-

nity agreement, and conceptuality and domain speci-

ﬁcity for the capturing of domain knowledge in form

of a conceptual model. In contrast to other conceptual

models used in computer science, such as UML dia-

grams of software systems or entity-relationship dia-

grams for data storage, the primary purpose of an on-

tology is to serve as a source of domain knowledge to

be queried by an information system that bases run-

time decisions on the respective answers.

Ofttimes, ontologies are visualised and thought of

as semantic networks that display interrelated concep-

tual nodes, as exemplarily depicted in Figure 1.

2.1 Essential Characteristics

Although there are various ontology languages with

different semantics for similar language constructs

next to a multitude of differing ontology formalisa-

tions, we can identify some essential characteristics

that seem to be common to all presentations of on-

tologies across different research areas.

• interrelation – Without any interconnection, the

plain nodes in a semantic network like the one

depicted in Figure 1 would rather be a list of

keywords. Relations between concepts allow for

such an interconnection, enriching the conceptual

nodes with structure. In the example, cars are re-

lated to engines by means of a

hasPart

connection.

• instantiation – An essential distinction is that

made between concrete individual objects of the

domain of interest and more general categories

that group together objects with some common

characteristics. Instantiation is the mechanism

to allow for this distinction by assigning individ-

ual objects to classes as their instances, such as

the speciﬁc Maserati in Figure1 is an instance

of the class of all cars. This mechanism can be

found across almost all forms of conceptual mod-

els, sometimes even in the form of metamodelling

(Motik, 2005) allowing for classes to be instances

of (meta-) classes themselves.

• subsumption – The most common way of inter-

linking general conceptual nodes is by subsump-

tion, expressing a

kindOf

-relationship that re-

ﬂects the notion of specialisation/generalisation.

In the example, cars are stated to be kinds of vehi-

cles, and thus, inherit their properties being their

specialisation. Subsumption is the mechanism be-

hind the feature of inheritance hierarchies preva-

lent in conceptual modelling.

• exclusion – A rather sophisticated means of repre-

sentation is to state “negative” knowledge in form

of class exclusion, preventingtwo general concep-

tual nodes from overlapping in their extensions.

In the example, cars are stated to be different from

ships, meaning that being a car excludes being a

ship. This form of negative knowledge is found

in rather expressive ontology languages that allow

for negation.

• axiomatisation – The interrelated conceptual

nodes as such are often not sufﬁcient to ex-

press rich knowledge – they need to take part in

complex statements about the domain of interest,

which accounts for the notion of axiomatisation.

Besides subsumption, exclusion or instantiation

as simple forms of axiomatisation, many ontology

languages allow for the formulation of more com-

plex statements in form of general axioms. Com-

plex axioms are typically neglected in the graph-

ical presentation of an ontology in lack of an ap-

propriate means for visualisation.

• attribution – Although not motivated by the un-

derlying logical formalisms for knowledge rep-

resentation, the inclusion of statements about

datatypes and their values are a common feature

in ontology languages, which accounts for the at-

tribution of conceptual nodes by strings, numbers,

etc. In the example, cars are attributed with a

number indicating their horsepower. Datatypes

and values are often an indispensable feature in

many applications of ontologies.

2.2 Model Design Goals

Our primary goal is to design a formal ontology

model that is on the one hand simple to suit a con-

cise presentation of ontologies, and on the other hand

rich enough to account for all the essential character-

istics identiﬁed above. It should allow for emphasis-

ing any of these characteristics while abstracting from

any of the others. In particular, it should be possible

to either take an axiomatic view on an ontology not

being concerned with structural aspects, or to take a

graph-like view not mixing in logic-based axiomati-

sation. Despite all abstraction, the model should be

KEOD 2009 - International Conference on Knowledge Engineering and Ontology Development

328

compatible to the common ontology languages and

formalisms for cases in which details of formal se-

mantics and other knowledge representation speciﬁcs

are to be discussed.

As the main use of such a unifying formal model

we see the possibility of presenting ontology-based

work in the course of Semantic Web research in a

uniform way across different areas. Once such a

model is established, however, it can facilitate more

far-ranging uniform handling of ontologies. On the

one hand, it can form the basis for a uniform graphical

visualisation of ontologies that accounts for the above

characteristics, e.g. in the widely accepted UML stan-

dard. On the other hand, it can serve as a structural

framework for a uniform programmatic access to on-

tologies in software systems in form of an API lay-

ered on top of existing ontology software back-ends,

e.g. in the widely used Java programming language.

3 A FORMAL MODEL FOR

ONTOLOGIES

In this section, we present our formal ontology model

as such, describing the constituents of an ontology as

well as a set of conditions that determine their char-

acteristics. Moreover, we report on some design deci-

sions that have inﬂuenced the model.

3.1 Constituents of an Ontology

An ontology O is a tuple

O = (S, A) (1)

of a signature S and a set of axioms A. The signature

comprises several sets

S = C∪ I∪ P (2)

of classes C, instances I and properties P. They are

further divided in sets

C = C ∪ D , I = I ∪ V , P = R ∪ T (3)

of concepts C and data types D , individuals I and

data values V , as well as relations R and attributes

T . We also call the elements of the signature the en-

tities of an ontology.

An axiom α ∈ A is an expression in a ﬁrst-order

logical language over the signature S with the follow-

ing restrictions:

• any entity E ∈ S occurs in some α ∈ A

• predicate symbols in α are taken from C ∪ P,

where symbols taken from C must be used as

unary predicates in α

• constant symbols in α are taken from I

• any t ∈ T must be used as a binary predicate t(i, v)

with i ∈ I and v ∈ V (or i, v ﬁrst-order variables)

Based on the essential characteristics from Sec-

tion 2, we identify some special types of axioms that

are common to most ontology languages and formal-

isations, for which we introduce a special notation in

Table 1. Their intuitive meaning is as follows.

• instantiation – An instantiation axiom assigns an

instance to a class.

• assertion – An assertion axiom assigns two in-

stances by means of a property.

• subsumption – A subsumption axiom for two

classes states that any instance of any one class

is also an instance of the other class, while for

two properties it states that any two instances con-

nected by the one property are also connected by

the other one.

• domain – A domain axiom for a property and

a class states that for any connection of two in-

stances by that property the source element is an

instance of the domain class.

• range – A range axiom for a property and a class

states that for any connection of two instances by

that property the target element is an instance of

the range class.

• disjointness – A disjointness axiom for two

classes states that no instance of the one class can

also be an instance of the other class, and thus, the

classes exclude each other.

3.2 Normative and Semantic Conditions

We deﬁne some conditions that can be employed to

further restrict the use of an ontology’s entities and

axioms. We divide them into normative conditions

for restricting the signature and semantic conditions

for restricting the semantics of axioms.

3.2.1 Normative Conditions

Some normative conditions that reﬂect basic aspects

of formalisms for conceptual models are listed in Ta-

ble 2. By separating axioms from entities, N

pre-

vents reiﬁcation of the form of axioms about entities

that are axioms themselves, such as statements about

triples in RDF (Klyne and Carroll, 2004). Moreover,

prevents any form of metamodelling by clearly

separating instances, classes and properties (no class

can be used in place of an instance). Furthermore, N

separates abstract domain objects from concrete data

A UNIFYING FORMAL ONTOLOGY MODEL - A Simple Formal Model for Unifying the Presentation of Ontologies in

Semantic Web Research

329

Table 1: Special axioms common in ontology formalisms.

Axiom Type Notation ﬁrst-order expression

instantiation α

∧

(i,C) [C(i)], i ∈ I,C ∈ C

assertion α

→

, p, i

) [p(i

, i

)], i

, i

∈ I, p ∈ P

subsumption α

△

, E

) [∀x : E

(x) → E

(x)], E

, E

∈ C∪ P

domain α

D→

(p, D) [∀x, y : p(x, y) → D(x)], p ∈ P, D ∈ C

range α

→R

(p, R) [∀x, y : p(x, y) → R(y)], p ∈ P, R ∈ C

disjointness α

⊕

) [∀x :C

(x) ∧C

(x) → ⊥], C

∈ C

Table 2: Normative conditions.

ID Name Condition(s)

non-reiﬁability A∩ S = ∅

non-metalayering a) I∩ C = ∅

b) I∩P = ∅

c) C∩ P = ∅

d) ∀r ∈ R : arity(r) > 1

data-separation a) C ∩ D = ∅

b) I ∩V = ∅

c) R ∩ T = ∅

relation-binarity ∀r ∈ R : arity(r) = 2

non-instantiation ∀α ∈ A :

for α = [. . . p(t) . . .],

if p ∈ C then t 6∈ I

values and their types, as e.g. done in OWL (Patel-

Schneider et al., 2004). By N

, only binary rela-

tions are allowed, and N

prevents the instantiation

of classes by restricting terms of class predicates in

axioms to not be constants.

3.2.2 Semantic Conditions

The notion of implicit knowledge associated with an

ontology O in terms of deduction is reﬂected in our

model by the symbol hAi, which denotes the deduc-

tive closure of the set A of axioms in O . Although dif-

ferent ontology languages and knowledge representa-

tion formalisms exhibit different semantics that yield

varying sets hAi for an ontology, we identify some se-

mantic conditions that hold for most of the commonly

used semantics, regardless of the underlying ontology

language or model. These conditions characterise the

“meaning” of the speciﬁc axioms outlined in Table 1

and are listed in the following.

: (superclass instantiation) if α

∧

(i,C

) ∈ hAi and

△

) ∈ hAi then α

∧

(i,C

) ∈ hAi

: (superproperty instantiation) if α

→

, p

, i

) ∈

hAi and α

△

, p

) ∈ hAi then α

→

, p

, i

) ∈ hAi

: (subsumption transitivity) if α

△

, E

) ∈ hAi

and α

△

, E

) ∈ hAi then α

△

, E

) ∈ hAi

: (property domain) if α

D→

(p, D) ∈ hAi and

→

, p, i

) ∈ hAi then α

∧

, D) ∈ hAi

: (property range) if α

→R

(p, R) ∈ hAi and

→

, p, i

) ∈ hAi then α

∧

, R) ∈ hAi

: (disjointness) if α

⊕

) ∈ hAi then

⊕

) ∈ hAi and there is no i ∈ I such that

both α

∧

(i,C

) ∈ hAi and α

∧

(i,C

) ∈ hAi

: (negativity) for C

∈ S there is a set of axioms

, . . . , α

such that α

⊕

) ∈ hA∪ {α

, . . . , α

3.3 Design Decisions

A major design goal for the presented ontology model

was to achieve simplicity for use cases in which an

overly formal presentation of an ontology would be

hindering. This simplicity is employed in a gradual

way providing several layers of detail that can be cho-

sen to present the constituents of an ontology at the

desired granularity.

At the top-most level (1), an essential distinction

is made between the vocabulary elements in the sig-

nature S and the actual statements that use these ele-

ments to express knowledge about the domain as ax-

ioms A. Hence, this level is appropriate when viewing

an ontology as a semantic vocabulary without further

distinction of the elements in S in terms of concepts,

relations, etc., or when working with the statements

about the domain in A as such without investigat-

ing their axiomatic structure or semantic properties.

This level already provides sufﬁcient detail for many

Semantic Web applications, such as semantically en-

hanced document retrieval where the elements of an

ontology’s signature are used for indexing. A partic-

ularly crucial design decision at this level is the clear

separation of axioms, which are often neglected in use

cases associated with light-weight semantics. In our

model, axioms are the statements about the domain,

whereas entities in the signature form the vocabulary

used within these statements, and the two are clearly

distinguished.

At the second level (2), the distinction between

classes C, instances I and properties P is introduced,

which is an essential feature in almost all forms of

conceptual knowledge representation ranging from

UML in software design to logically expressive on-

tology languages like OWL. Hence, this level intro-

duces the notions of instantiation and interrelation,

and is appropriate when explicitly working with the

KEOD 2009 - International Conference on Knowledge Engineering and Ontology Development

330

constituents of an ontology’s signature as classes, in-

stances and properties. This level will be sufﬁcient

for most Semantic Web applications that exploit the

structural features of an ontology but do not rely on

the formal semantics that underlies the left out ax-

iomatic details of the elements in A. Examples are

various works on ontology mapping, Web 2.0 or many

approaches that take a graph-oriented view on ontolo-

gies as interconnected classes and their instances.

At the third level (3), the participants of instanti-

ation, namely classes and instances, are further split,

and abstract concepts C are distinguished from data

types D like string or integer, while abstract individ-

ual objects I of the domain are separated from con-

crete data values V . Accordingly, properties are split

into relations R , which involve abstract domain ob-

jects only, and attributes T , which involve also data

values. This design decision reﬂects the natural sep-

aration of abstract domain objects from attributes of

such objects, which are merely data values attached to

them, and thus accounts for attribution.

In this sense,

individuals serve as instances of concepts, while data

values serve as instances of data types. Hence, this

level is appropriate when working with the explicit

distinction between abstract objects of the domain and

concrete data values, which is often the case in e.g.

information integration scenarios where an ontology

serves as a unifying schema for databases.

At an even ﬁner-grained level, touching the

grounds of formal semantics attached to ontology

languages and deduction mechanisms, our ontology

model renders the statements about the domain in A

as expressions of a ﬁrst-order logical language that

employs the elements of an ontology’s signature S as

the symbols for predicates and constants. Although

we do not aim at providing a comprehensive ontol-

ogy language with formal syntax and semantics, we

make use of the well established ﬁrst-order logic no-

tation as a unifying syntactical framework for axioms

in an ontology.

In this notation the statements of

most ontology languages can be expressed and it nat-

urally accounts for the notions of class, relation and

instance by means of unary predicates, binary predi-

cates and constants. Furthermore, we provide a min-

imal set of semantic conditions that formalise the es-

sential characteristics shared by most knowledge rep-

This distinction is also made in entity-relationship

modelling in the ﬁeld of databases, in object-oriented soft-

ware design, and in ontology languages like OWL, where

object properties are separated from datatype properties.

First-order predicate logic (FOL) has been used as a

unifying umbrella formalism for many different forms of

knowledge representation, such as description logics or

logic programming, which can be understood as syntactic

and/or semantic deviations of FOL.

resentation formalisms and ontology languages in our

model, such as the interplay between instantiation and

subclassing (S

), the restriction of a relation’s

domain and range classes (S

), the symmetric ex-

clusion of class extensions (S

) and the ability of de-

riving class disjointness (S

4 CONFORMANCE WITH

SEMANTIC WEB RESEARCH

In this section, we demonstrate the compatibility and

conformance of our proposed ontology model to ex-

isting notions of an ontology in various areas of Se-

mantic Web research, such as ontology language stan-

dards, light-weight semantics or ontology mapping.

We do so by specifying mappings from different on-

tology formalisations used in those areas to our sim-

ple ontology model.

4.1 Mapping from OWL

The Web Ontology Language(OWL)(Patel-Schneider

et al., 2004) is currently the most prominent of the

W3C standard languages for expressing ontologies

on the web. It provides for expressive knowledge

representation based on the underlying description

logic (DL) formalism (Baader et al., 2003).

In Table 3, we specify how the various elements

of an OWL ontology can be mapped to the signature

and axioms of our ontology model, where we use the

DL style notation for presenting the OWL elements.

An OWL ontology is seen as a DL knowledge base

KB that consists of DL axioms, while N

, N

are sets of named entities (individuals, classes, ob-

ject properties and datatype properties) used within

these axioms and D

are concrete domains that repre-

sent datatypes in OWL.

By means of the syntactical ﬁrst-order logic um-

brella that we use for axioms in our model, the DL

statements directly map to the axioms in A, since

DLs are fragments of ﬁrst-order predicate logic. For

named classes C

(i)

∈ N

, individuals a

(i)

∈ N

and

properties r

(i)

∈ N

, we specify some particular DL

axioms in terms of the notation introduced in Table 1.

The imposed conditions N

- N

render OWL as

a rather strictly deﬁned ontology language with only

binary relations and a clear separation of the different

types of entities that prevents any form of metamod-

elling or reiﬁcation. Moreover, the conditions S

ensure all the semantic properties common in rich

knowledge representation formalisms, such as transi-

tivity of or propagation of instances along subsump-

tion hierarchies as well as negativity.

A UNIFYING FORMAL ONTOLOGY MODEL - A Simple Formal Model for Unifying the Presentation of Ontologies in

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331

Table 3: Mapping OWL.

OWL element signature element

, . . . , D

} D

∪ · ·· ∪ D

KB A

DL expression axiom

C(a) ∈ KB α

∧

(i,C) ∈ A

r(a

, a

) ∈ KB α

→

, r, a

) ∈ A

⊑ C

∈ KB α

△

) ∈ A

⊑ r

∈ KB α

△

, r

) ∈ A

∃r.⊤ ⊑ C ∈ KB α

D→

(r,C) ∈ A

⊤ ⊑ ∀r.C ∈ KB α

→R

(r,C) ∈ A

⊓C

⊑ ⊥ ∈ KB α

⊕

) ∈ A

conditions: N

- N

, S

- S

4.2 Mapping from RDFS

The Resource Description Framework RDF (Klyne

and Carroll, 2004) with its schema extension RDFS

(Brickley and Guha, 2004) is the prevalent standard

for many applications in the area of linked open data.

For an RDF graph G with triples hr

, r

i, Table 4

speciﬁes the mapping of an RDFS ontology repre-

sented by G to the simple ontology model, where R

is the set of resources that occur in G and r

∈ R are

speciﬁc resources.

Table 4: Mapping RDFS.

RDFS element signature element

| hr

, rdf:type, r

i ∈ G∨ I

, r

i ∈ G, r

6∈ Voc

RDF

∨

, r

i ∈ G, r

6∈ Voc

RDF

}

| hr

, rdf:type, r

i ∈ G∨ C

, rdf:type, rdfs:Classi ∈ G∨

, rdfs:subClassOf, r

i ∈ G∨

, rdfs:subClassOf, r

i ∈ G∨

, rdfs:domain, r

i ∈ G∨

, rdfs:range, r

i ∈ G}

| hr

, r

i ∈ G∨ P

, rdf:type, rdfs:Propertyi ∈ G∨

, rdfs:subPropertyOf, r

i ∈ G∨

, rdfs:subPropertyOf, r

i ∈ G∨

, rdfs:domain, r

i ∈ G∨

, rdfs:range, r

i ∈ G}

{rdfs:Literal} D

< string > V

RDF(S) triple axiom

, rdf:type, r

i ∈ G α

∧

, r

) ∈ A

, r

i ∈ G, r

6∈ Voc

RDF

→

, r

) ∈ A

, rdfs:subClassOf, r

i ∈ G α

△

, r

) ∈ A

, rdfs:subPropertyOf, r

i ∈ G α

△

, r

) ∈ A

, rdfs:domain, r

i ∈ G α

D→

, r

) ∈ A

, rdfs:range, r

i ∈ G α

→R

, r

) ∈ A

conditions: N

, S

- S

We map the triples to axioms in a way such that

plain triples not containing any predeﬁned RDF(S)

vocabulary terms are interpreted as assertion axioms

connecting instances. Special triples that contain

RDF(S) vocabulary like rdf:type, rdf:subClassOf, etc., map

to instantiation axioms, subclass axioms, etc.

Of the normative conditions, only N

is imposed,

which renders RDFS as a rather unconstraint lan-

guage that allows for reiﬁcation and metamodelling.

The imposed semantic conditions S

- S

reﬂect some

semantic features of RDFS, such as the instance-class

connection along subsumption hierarchies and do-

main/range properties. Moreover, the lack of condi-

tions S

renders RDFS as a formalism that does

not have the expressivity for negation to state or de-

tect logical inconsistencies due to class disjointness.

4.3 Mapping from SKOS

The Simple Knowledge Organization System SKOS

(Miles and Bechhofer, 2008) is a prominent light-

weight representation formalism in the area of

Web 2.0 and light-weight semantics based on

RDF(S). In Table 5, we specify a mapping from

SKOS concept schemes to our simple ontology

model. Similar to the mapping from RDFS, we base

the SKOS mapping on an RDF graph G that contains

the respective concept scheme, while we take into ac-

count speciﬁc SKOS vocabulary.

Table 5: Mapping SKOS.

SKOS element signature element

{r | hr, rdf:type, skos:Concepti ∈ G} I = C

{string} D

{skos:related} R = {r

rel

}

{skos:XLabel, skos:Note} T = {t

xLab

Note

}

< string > V

SKOS RDF triple axiom

, skos:broader, r

i ∈ G α

△

, r

) ∈ A

, skos:narrower, r

i ∈ G α

△

, r

) ∈ A

, skos:related, r

i ∈ G α

→

, r

rel

, r

) ∈ A

hr, skos:XLabel, vi ∈ G α

→

(r, t

xLab

, v) ∈ A

hr, skos:note, vi ∈ G α

→

(r, t

Note

, v) ∈ A

conditions: N

, N

Since in SKOS classes are not explicitly distin-

guished from instances, we let the two respective

sets coincide and include all elements from a con-

cept scheme. This allows for using SKOS concepts

both in place of classes to build class hierarchies and

in place of instances to interlink them. As there are

no custom relations or attributes in SKOS, the respec-

tive sets comprise the speciﬁc SKOS vocabulary only.

Subclass axioms reﬂect the broader/narrower hierar-

chy essential to SKOS, while other axioms represent

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relatedness of concepts and their string attributes with

the labels, notes, etc. predeﬁned in SKOS.

Besides the imposed normative conditions N

for

preventing reiﬁcation, N

for separation of data val-

ues and their types and N

for the restriction to binary

predicates, condition N

supports the class/instance

coincidence mentioned above by preventing concepts

from being used to predicate about individuals. On

the semantic side, SKOS does not support rich deduc-

tive features for its predeﬁned vocabulary, which is

reﬂected by not imposing any semantic conditions.

4.4 Mapping to OA Model

Another area in Semantic Web research is concerned

with the mapping and alignment of ontologies (OA),

where similar entities across different ontologies are

to be identiﬁed. We specify a mapping from the

formalisation of ontologies in this area, taken from

(Euzenat and Shvaiko, 2007), to our simple ontology

model. According to this formalisation, an ontology

is characterised by classes

, instances

, relations

, datatypes

and their values

, while speciﬁc re-

lation over these sets denoted by ≤, ⊥, ε, = express

subsumption, exclusion, instantiation and assertion.

Table 6: Mapping OA Model.

OA model element signature element

OM model statement axiom

) ∈≤ ∩

△

) ∈ A

, p

) ∈≤ ∩

△

, p

) ∈ A

) ∈ ⊥ ∩

⊕

) ∈ A

, p

) ∈ ⊥ ∩

[∀x, y : p

(x, y) → ¬p

(x, y)] ∈ A

(i,C) ∈ ε α

∧

(i,C) ∈ A

, r, i

) ∈= ∩

→

, r, i

) ∈ A

conditions: N

, N

, S

- S

The basic elements of the ontology formalisation

from the alignment and mapping area are very similar

to the signature in our model and map almost one-to-

one, while the various axioms are determined by the

relations ≤, ⊥, ε, =.

Except for the clear separation between object

properties and datatype properties, all the normative

conditions N

, N

can be imposed, together with

the expressive set of semantic conditions S

- S

as in

the OWL case, which reﬂects the fact that this formal-

isation builds on a similar model-theoretic semantics.

In Table 5, skos:XLabel stands for the various label ele-

ments and skos:note represents the different documentation

properties deﬁned in SKOS.

5 UML VISUALISATION

Next to their formal notation, the presentation of on-

tologies in research works often also requires the

graphical illustration of conceptual models that dis-

play an ontology’s constituents in an intuitively gras-

pable way, for example, to give an overview on a

back-bone taxonomy containing the most essential

concepts. The Uniﬁed Modelling Language UML

well established in the software development com-

munity, is a prevalent instrument for displaying con-

ceptual models of all kinds, in particular by means of

UML class diagrams, and we propose to use it for this

purpose in connection to our formal model.

Applied in the standard way, however, UML class

diagrams offer some freedom in the use of their man-

ifold elements, which results in a multitude of dif-

ferent styles of diagrams when used for ontologies,

similar to the many different formal ontology models

found in research works. This calls for a well-deﬁned

uniform way of using the graphical UML elements

for displaying at least the essential characteristics of

ontologies common to most research contexts. As

an additional requirement, the graphical visualisation

should be inline with the formal notation and reﬂect

an ontology’s formal structure as close as possible,

whereas in many presentations ontology diagrams are

rather vague with no clear interpretation of graph ele-

ments in relation to the underlying formal model.

To provide a uniform way of displaying ontolo-

gies in UML, we propose a UML metamodel exten-

sion that guides the use of UML class diagrams for

graphical presentation of ontologies in accordance to

our formal ontology model, as depicted in Figure 2. It

deﬁnes the entities in S of an ontology O as extensions

of predeﬁned UML metaclasses, such as UML::Class or

UML::Association in form of new UML stereotypes for

classes, instances and properties.

The axioms A are

accounted for by an abstract stereotype with special-

isations for the different axiom types α

△

, α

⊕

, α

∧

D→

/α

→R

and α

→

The connections between stereo-

types for axioms and entities indicate how an axiom is

expressed in terms of UML relations. For example, an

axiom of type α

∧

is a UML association that involves

one instance and one class.

In summary, the UML metamodel extension pro-

vides means to visualise the essential characteristics

of ontologies in a way consistent to our formal model.

http://www.uml.org/#UML2.0

For simplicity, we left out stereotypes for the more spe-

ciﬁc concepts, individuals, data values, etc.

The axiom types for domain and range restriction are

combined in the graphical model with the idea that a di-

rected association between classes expresses both.

A UNIFYING FORMAL ONTOLOGY MODEL - A Simple Formal Model for Unifying the Presentation of Ontologies in

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333

«stereotype»

Class

«metaclass»

UML::Class

«metaclass»

UML::Instance

«metaclass»

UML::Association

«metaclass»

UML::Generalization

«stereotype»

Instance

«stereotype»

Property

«stereotype»

Axiom

«stereotype»

«metaclass»

UML::Classifier

D R

Figure 2: UML metamodel for ontology visualisation.

In this way, we can beneﬁt from the existing tool

support for the visual creation of graphical ontology

models by means of UML editors, which also enables

the possibility for the generation of code in various

ontology languages, e.g. through EMF

transforma-

tions based on the mappings presented in Section 4.

6 AN ONTOLOGY API

Our formal ontology model can also form the basis for

a uniﬁed programmatic access to ontologies by means

of a software API that is layered on top of existing

ontology back-end systems. This allows information

systems to take a simpliﬁed view on ontologies on the

software side.

In Figure 3, we sketch an architecture for such an

API and its implementation in form of a UML dia-

gram. An ontology and its axioms can be accessed by

OOntologyImpl

OSignature

OAxiomImpl

OEntity

OOntology

OAxiom

Figure 3: An ontology software API.

respective interfaces whose implementing classes es-

tablish the connection to the ontology’s signature and

entities. For sake of simplicity, we left out specialised

types for speciﬁc axioms and entities.

Through implementations for various ontology

back-end systems, such as e.g. the OWL-API (Hor-

ridge et al., 2007), an ontology-based application can

access and manipulate ontologies in an easy way, ab-

stracting from the details of the underlying ontology

language layer if desired.

http://www.eclipse.org/modeling/emf/

7 CONCLUSIONS

We have proposed a unifying formal model for on-

tologies that allows for both simplicity in presenta-

tion and representation of the essential characteristics

of an ontology. We have demonstrated compatibility

to SW research works by having provided mappings

to various existing ontology languages and formali-

sations. Moreover, we have sketched the further use

of our model as a basis for graphical visualisation in

terms of UML and for a simpliﬁed programmatic ac-

cess to ontologies via a speciﬁc API.

We argue that our unifying model can help to me-

diate between different strands of SW research, pro-

viding a common view on ontologies that captures

their essentials while abstracting from details of spe-

ciﬁc languages. We aim to work towards a more thor-

ough system of normativeand semantic conditions, as

well as the realisation of UML-based ontology visu-

alisation and a unifying software API with implemen-

tations for various ontology frameworks.

ACKNOWLEDGEMENTS

This work is partially supported by the German Fed-

eral Ministry of Economics (BMWi) under the project

THESEUS (number 01MQ07019).

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