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Representing n-ary relations in the Semantic Web

Marco Giunti, Giuseppe Sergioli, Giuliano Vivanet, and Simone Pinna

Università di Cagliari

ALOPHIS (Applied Logic Philosophy and HIstory of Science)

Dipartimento di Pedagogia Psicologia Filosoﬁa

via Is Mirrionis 1, 09123 Cagliari, Italy

giunti@unica.it

giuseppe.sergioli@gmail.com

giuliano.vivanet@unica.it

simonepinna@hotmail.it

Abstract. Knowledge representation is a central issue for Artiﬁcial In-

telligence and the Semantic Web. In particular, the problem of represent-

ing n-ary relations in RDF-based languages such as RDFS or OWL by

no means is an obvious one. With respect to previous attempts, we show

why the solutions proposed by the well known W3C Working Group

Note on n-ary relations are not satisfactory on several scores. We then

present our abstract model for representing n-ary relations as directed

labeled graphs, and we show how this model gives rise to a new ontolog-

ical pattern (parametric pattern) for the representation of such relations

in the Semantic Web. To this end, we deﬁne PROL (Parametric Rela-

tional Ontology Language). PROL is an ontological language designed

to express any n-ary fact as a parametric pattern, which turns out to be

a special RDF graph. The vocabulary of PROL is deﬁned by a simple

RDFS ontology. We argue that the parametric pattern may be particu-

larly beneﬁcial in the context of the Semantic Web, in virtue of its high

expressive power, technical simplicity, and faithful meaning rendition.

Examples are also provided.

Keywords: Knowledge representation, semantic net, Semantic Web,

RDF, RDFS, n-ary relation, formal ontology, ontological language, on-

tological pattern, relation reiﬁcation, reiﬁed relation

1 Introduction

The development of the Semantic Web originated from the view that the whole

complex of our knowledge may eventually be translated into a machine-readable

language, which will be interpreted and understood by artiﬁcial agents, and will

thus promote cooperation and exchange between humans and machines (Berners

Lee, Hendler and Lassila [1]). A fundamental requirement for this language is

that the complex of its statements constitutes a net (in mathematical terms, a

directed, labeled graph), in which the semantic relations between statements are

made explicit.

2 Marco Giunti et al.

The idea that the entire human knowledge may ultimately form a complex

semantic net is by no means new. For instance, consider Hempel’s metaphor of

scientiﬁc theories as nets ([13], p. 36), or Quine’s holistic conception of meaning

([19], Sect. VII). Nevertheless, in the context of the Semantic Web this old idea

gains a new strength as, for the ﬁrst time, it is supported by a set of information

technologies, which may very well transform that mere metaphor into reality.

RDF is the declarative language that, together with its ontological extension

RDFS (RDF Schema) and the more powerful OWL (an ontological language

based on description logics and compatible with RDF), may allow us to represent

an arbitrary knowledge base as a directed labeled graph. In fact, according to the

W3C speciﬁcations of RDF, any set of its statements can be formally identiﬁed

with an appropriate directed labeled graph. Each node of the graph is an IRI

(Internationalized Resource Identiﬁer) of a resource. A resource is an entity of

any kind (document, thing, event, concept, etc.), while its IRI is an appropriate

string of Unicode characters that globally and univocally identiﬁes the resource.

Each arrow of the graph is labeled by an IRI that identiﬁes a two-place relation

(called a “property” in RDF, RDFS, and OWL). Such a relation is intended to

hold for the ordered pair of resources that corresponds to the source and target

node of the arrow.

However, the expressive power of RDF is quite limited, for the following

reasons. (i) RDF does not have any means to express the negation of a sentence;

(ii) RDF can only express the conjunction of two or more atomic sentences, but it

does not have any inbuilt capacities to express the other connectives (disjunction,

implication, double implication); (iii) RDF has very limited facilities (called

“blank nodes”) to express quantiﬁed statements. Blank nodes only allow for a

purely existential quantiﬁcation of the conjunction of an arbitrary number of

atomic sentences; (iv) all predicates of RDF have two places. This means that

RDF can directly express only those statements that involve two-place relations.

Statements that involve unary relations (John is tall) or relations with three or

more places (John gives a rose to Mary) can only be indirectly expressed in RDF,

by ﬁrst translating such relations into convenient binary ones. As far as unary

relations are concerned, RDF provides inbuilt facilities for their translation (RDF

classes and the special property rdf:type). Nevertheless, for n-ary relations with

n≥3there are no approved W3C standards (known as “Recommendations”) for

their expression in either RDF, RDFS, or OWL. Even though, so far, the W3C

has not released approved standards for representing arbitrary n-ary relations,

it has nonetheless suggested a number of ontological patterns. These suggestions

have the status of a “Working Group Note” (W3C [23]), but we will show in

detail (see Sects. 3 and 4.2) why they cannot be accepted as deﬁnitive solutions.

The severe expressive limits of RDF are partially overcome by its ontological

extension RDFS and, to a larger extent, by OWL. However, even OWL 2 (the

most recent version of OWL) does not reach the full expressive power of a ﬁrst

order language. An unfortunate consequence of this fact is that, at the moment,

huge portions of our knowledge cannot be made available on the Semantic Web

as linked data, not even in principle.

Representing n-ary relations 3

The main aim of this work is to propose a new ontological pattern for repre-

senting n-ary relations, which we call parametric pattern. In Sect. 2, we introduce

and deﬁne the general notion of a n-ary relation. In Sect. 3, we analyze and dis-

cuss the ontological patterns proposed by the W3C. In Sect. 4, we develop an

abstract graph model (ﬁrst presented in Giunti [9]) for the representation of n-

ary relations by means of binary ones. This abstract graph model constitutes the

theoretical basis of the new parametric pattern. Finally, in Sect. 5, we show how

it is possible to transform our graph model into an actual ontological pattern,

which allows us to represent an arbitrary n-ary relation by means compatible

with RDF.

To this end, we deﬁne PROL (Parametric Relational Ontology Language).

PROL is a simple ontological language designed to express any fact that in-

volves a n-ary relation (i.e., any n-ary fact) as a parametric pattern, which is

indeed a special RDF graph. The vocabulary of PROL includes just 6 terms (2

RDF classes and 4 RDF properties) deﬁned by a simple RDFS ontology. Two

terms (prol:type, prol:next) serve to represent any n-ary fact as a parametric

pattern. The remaining four terms (prol:Relation, prol:Domain, prol:hasPlaces,

prol:represents) serve to express the ontology that (a) deﬁnes the n-ary relations

involved in the facts to be represented, and (b) allows for the correct detection

and interpretation of the representing parametric patterns.

We maintain that, to date, our proposal is one of the simplest and most

eﬀective solutions to the problem of representing n-ary relations in the Semantic

Web. However, the interest of this proposal is not limited to the solution of

this particular problem, namely, to remove the speciﬁc expressive limit of RDF

described in (iv) above. In a wider perspective, we maintain that the parametric

pattern may be a crucial ﬁrst step to tackle and solve also the other three

expressive limits of RDF, or related ontological languages (see Sect. 6). As soon

as these further developments give rise to widespread and shared standards, we

may ﬁnally obtain a new ontological language compatible with RDF that, on

the one side, will be as expressive as any ﬁrst order language and, on the other

one, will also be able (like the present RDF) to connect all its statements in a

graph. This will produce a very powerful semantic net, which in principle will

be able to realize the original vision of the Semantic Web.

In addition, our proposal can be seen as an application of model based reason-

ing (Magnani, Nersessian and Thagard [15]; Magnani and Bertolotti [16]) to the

logical analysis of current practices of knowledge representation in the Semantic

Web and, speciﬁcally, with respect to the role of diagrammatic representation

and the ontology of models.

2 N-ary relations

According to the deﬁnition proposed by Gruber, an ontology

[. . .]deﬁnes a set of representational primitives with which to model a

domain of knowledge or discourse. The representational primitives are

4 Marco Giunti et al.

typically classes (or sets), attributes (or properties), and relationships

(or relations among class members). (Gruber [12], p. 1963)

In the relevant literature, however, the meaning attributed to each of the above

terms (classes or sets, attributes or properties, relations) is not always univocal.

For example, let us consider the following statement: “this apple is red”. Even

though being red is usually conceived as a property of the apple or, in other

words, as one of its attributes, this statement is interpreted in RDF as involving

a relation (rdf:type) between the apple and the class of red things. Also note

that, in RDF, the term “property” is not a synonym of “attribute”, but rather of

“binary relation” (W3C [24], Sect. 3).

There are also cases in which an attribute is intended as a modiﬁer of a binary

relation (i.e., a property in RDF sense). For instance, one of the W3C use cases

discussed in Sect. 3 is “Christine has breast tumor with high probability”, where

the expression “with high probability” is meant as an attribute which modiﬁes

the binary relation of Christine’s having breast tumor.

Furthermore, in RDF, classes are not intended as just sets of elements, but

rather as concepts or classiﬁcations (intensions) whose extensions are sets (W3C

[22], Sect. 1.1). For example, the Math exam candidates and the English exam

candidates are two diﬀerent classes, but their elements may be exactly the same.

Analogously, RDF distinguishes between a property, namely, a binary relation

intensionally deﬁned, and its extension, i.e., the set of ordered pairs for which

the given relation holds.

In this paper, we are mainly concerned with the general concept of a relation

and, in particular, with the problem of representing n-ary (n≥3) relations

by only employing binary ones. As suggested by common usage, a relation is

a connection that does or does not hold for a certain number of individuals.

For instance, the relation less than does hold for the numbers one and three,

but it does not for three and two. This simple example already shows that

order is a fundamental feature of the general concept of a relation. In general,

a relation holds, or does not, for a certain number of individuals in a given

order. Relations that hold irrespectively of the order of the individuals involved

are called symmetric and they are in fact a proper subclass of the relations in

general.

Another fundamental feature of a relation is its arity, which indicates the

number of individuals involved.1In the previous example we have considered

the relation ()less than(), which holds for two individuals in a given order. In

this case, the arity of the relation is 2, and it is called a binary relation. The

elements of this relation are all the ordered pairs of individuals for which the

relation holds. Consider now the statement “two is between one and three”. Here

the individuals involved are three, so that the relation ()is between()and() has

arity 3. In general, a relation with arity n≥1holds, or does not, for exactly

1The individuals involved may only be numerically distinct, that is to say, the same

individual may be involved more than once in the same relation. The typical example

is the identity relation, which only holds for any pair of individuals whose ﬁrst and

second member are the same.

Representing n-ary relations 5

nindividuals in a given order, and its elements are all the ordered n-tuples

of individuals for which the relation holds. A n-ary relation is a relation whose

arity is n. In this paper, we do not consider relations whose arity is not deﬁnitely

ﬁxed.2

To formally express the fact that a n-ary relation Rnholds for nordered

individuals a1, ... , an, we write: Rn(a1, ... , an). We adopt the usual convention

that the individuals involved in the relation are indicated by lowercase letters,

while predicates are denoted by uppercase letters combined with a superscript

that indicates the arity of the relation for which the predicate stands. As usual,

we intend that the arity of a predicate is equal to the arity of the relation for

which it stands.

In set theory and formal semantics it is customary to identify a n-ary relation

with the class of its elements, that is, with the class of all ordered n-tuples

for which the relation holds. We adopt here a similar identiﬁcation. However,

in order to be consistent with RDF conception of classes, we intend classes

intensionally. This means that we take any n-ary relation to be identical to a

class of ordered n-tuples, but two n-ary relations whose elements are exactly

the same not necessarily are the same class (in other terms, the extensionality

principle does not hold).

In RDFS, a domain of a property Pis deﬁned as any class C1such that,

∀x1∀x2, if x1P x2, then x1is a C1. Analogously, a range of Pis any class C2

such that, ∀x1∀x2, if x1P x2, then x2is a C2. We generalize these notions to

any n-ary relation as follows. Let Rnbe an arbitrary n-ary relation, and Cibe

a class (1≤i≤n). Ciis an i-th domain of Rniﬀ ∀x1...∀xn, if Rn(x1, ... , xn),

then xiis a Ci.

According to RDFS, a property P2is a subproperty of a property P1just

in case ∀x1∀x2, if x1P2x2, then x1P1x2. We generalize this concept to n-ary

relations as follows. With the notation Z[1,m]we mean the set of all positive

integers ksuch that 1≤k≤m, where m∈Z. Let Rnbe a n-ary relation, and

1≤m≤n; we say that pis a choice of mplaces of Rniﬀ pis an injective

function from Z[1,m]to Z[1,n]. Let Rmbe a m-ary relation and Snbe a n-ary

relation, where 1≤m≤n; let pbe a choice of mplaces of Sn;Snis a subrelation

of Rmwith respect to piﬀ ∀x1...∀xn, if Sn(x1, ... , xn), then Rm(xp(1), ... , xp(m)).

The logical structure of relations described above allows us to formally rep-

resent many facts usually expressed by statements of the natural language. For

example, the statement “Mario buys a jacket” can be analyzed as “(Mario) buys

a (jacket)”, where the binary relation represented by “()buys a()” holds for the

individuals denoted by “Mario” and “jacket”, in this order. We can thus intro-

duce the predicate “B2” to stand for the binary relation ()buys a(), and the

two constants “m” and “ j” for Mario and the bought jacket, namely, the two

individuals involved in the relation ()buys a(). This leads to formally represent

the unanalyzed statement “Mario buys a jacket” as “B2(m, j)”, which in fact is

just an abbreviation for the analyzed form “(Mario) buys a (jacket)”.

2See Oliver and Smiley [18] for so called “multigrade predicates”.

6 Marco Giunti et al.

These analytical tools have been traditionally used for formalizing the argu-

mentative structure of natural language. Consider the statement “Giuliano hits

Giuseppe hard”. For the sake of reconstructing an argument in which this state-

ment might be involved, the most natural analysis would probably be “(Giu-

liano) hits (Giuseppe)”, where the further information brought by the adverb

“hard”, which modiﬁes the meaning of the binary predicate “()hits()”, would be

discarded.

However, an encoding in a Semantic Web context would not necessarily be

aimed at the argumentative structure of natural language. Hence, in this wider

context, a ﬁner grained analysis might be needed. In the previous example,

it might be useful to consider three individuals: Giuliano, Giuseppe, and the

force with which Giuliano hits Giuseppe (not so hard, hard, really hard, etc.).

The term “hard” now stands for a further individual involved in the relation.

Consequently, the statement “Giuliano hits Giuseppe hard” should be analyzed

as “(Giuliano) hits (Giuseppe) (hard)”, which would involve the three ordered

individuals Giuliano, Giuseppe, hard, and the ternary relation ()hits()(). Note

that, according to our deﬁnition of a subrelation, the ternary relation ()hits()()

is a subrelation of the binary relation ()hits(), with respect to the choice pof

two places of ()hits()() such that p(1) = 1 and p(2) = 2.

In this paper, a relation is intended in a wider sense than the traditional one

rooted in argumentation theory. A relation is thought as a connection between

nindividuals in a given order, but some of these individuals may result from a

process of reiﬁcation (Brachman and Levesque [3], p. 41) similar to the one just

considered.

As noted above, in RDF, as well as in RDFS and OWL, properties are in

fact binary relations. However, in some cases, we need to represent relations that

involve more than two individuals, namely, relations of arity n≥3, as seen in

the previous example. In the literature, we can ﬁnd several diﬀerent approaches

to the treatment of relations of this kind. Many of them have originated from the

ﬁeld of Artiﬁcial Intelligence and the related problems of knowledge represen-

tation, and may be traced back to the introduction of semantic nets as models

for representing generic knowledge domains. Semantic nets are interconnected

systems of nodes and arrows, which can be thought as special kinds of directed

and labeled graphs (Brachman [2]; Sowa [20]).

The problem of representing n-ary relations with arity n≥3was explicitly

treated in this context by Deliyanni and Kowalski, who proposed to represent

the statement “John gives the book to Mary” as a special semantic net formed

by ﬁve nodes and four arrows ([4], p. 184). The general model they employed for

representing an arbitrary n-ary relation ([4], p. 186) by means of binary ones is

today known as relation reiﬁcation (Galton [5]; Gangemi [6]; Hobbs [14]; Masolo

[17]; Welty and Fikes [21]).

The basic idea of relation reiﬁcation consists in thinking of each n-tuple

of individuals for which a n-ary relation holds as an instance of a specially

introduced class, which is thought as a reiﬁcation of the relation itself, and can

thus be called a relation-class. The instance of the relation-class is then connected

Representing n-ary relations 7

to each of the nindividuals by means of nbinary relations, sometimes called

roles, which are speciﬁcally introduced to collectively approximate the meaning

of the original n-ary relation. In Sect. 4.2 we highlight several weaknesses of this

approach.

As mentioned before, in the literature we can ﬁnd a number of diﬀerent ap-

proaches to the representation of n-ary relations. For a good annotated bibliog-

raphy and a comparison of seven ontological patterns, see Gangemi and Presutti

([7], [8]). In the next Sect. 3, we analyze probably the most inﬂuential ontological

patterns based on the relation reiﬁcation model, which can be found in the well

known Working Group Note on n-ary relations (W3C [23]).

3 Two ontological patterns based on relation reiﬁcation:

The W3C Working Group proposal

The Working Group Note on n-ary relations (W3C [23]) considers four use case

examples in which n-ary (n≥3) relations are involved:

1. Christine has breast tumor with high probability.

2. Steve has temperature, which is high, but falling.

3. John buys a “Lenny the Lion” book from books.example.com for $15 as a

birthday gift.

4. United Airlines ﬂight 1377 visits the following airports: LAX, DFW, and

JFK.

In Example 1, the meaning of the binary predicate “()has()” is modiﬁed by

the attribute “high”, which speciﬁes a qualitative probability value. Besides the

two individuals Christine and breast tumor, for which the binary relation ()has()

holds, the attribute “high” is interpreted as standing for a third individual. There-

fore, the statement “Christine has breast tumor with high probability” turns out

to be analyzed as “(Christine) has (breast tumor) with (high) probability”, which

involves the ternary relation ()has()with()probability.

In Example 2, the meaning of the unary predicate “()has temperature” is

modiﬁed by two attributes, “high” and “falling”, which respectively specify a

qualitative temperature value and its trend. Besides the individual Steve, for

which the unary relation ()has temperature holds, the two attributes “high” and

“falling” are then interpreted as two further individuals. Therefore, the statement

“Steve has temperature, which is high, but falling” is in fact analyzed as “(Steve)

has temperature, which is (high), but (falling)”, which involves the ternary rela-

tion ()has temperature, which is(), but().3

3One might wonder why not considering instead the binary predicate “()has()” and

the further individual constant “temperature”, so that, in the end, the unanalyzed

statement would be analyzed as involving the 4-ary relation ()has(), which is(),

but(). According to the W3C Working Group Note, however, this alternative analysis

is excluded because: “In most intended interpretations, this instance of a relation

cannot be viewed as an instance of a binary relation with additional attributes

attached to it.” (W3C [23], Sect. 3.2.2).

8 Marco Giunti et al.

Fig. 1. Ontological pattern 1 – subject version – for: Christine has breast tumor with

high probability.

Fig. 2. Ontological pattern 1 – subject version – for: Steve has temperature, which is

high, but falling.

Representing n-ary relations 9

In Example 3, it is clear from the start that a n-ary (n≥3) relation holds

for the ﬁve individuals John, “Lenny the Lion” book, books.example.com, $15,

birthday gift. Therefore, the statement “John buys a ‘Lenny the Lion’ book

from books.example.com for $15 as a birthday gift” is analyzed as “(John) buys

a (‘Lenny the Lion’ book) from (books.example.com) for ($15) as a (birthday

gift)”, which involves the 5-ary relation ()buys a()from()for()as a().

Fig. 3. Ontological pattern 1 – no subject version – for: John buys a “Lenny the Lion”

book from books.example.com for $15 as a birthday gift.

In Example 4, “[...] the relation holds between the ﬂight and the airports

it visits, in the order of the arrival of the aircraft at each airport in turn.”

(W3C [23], Sect. 3.3). Therefore, the statement “United Airlines ﬂight 1377 visits

the following airports: LAX, DFW, and JFK” is in fact analyzed as “(United

Airlines ﬂight 1377) visits the following airports: (LAX), (DFW), and (JFK)”,

which involves the 4-ary relation ()visits the following airports:(),(), and().

Fig. 4. Ontological pattern 2 – subject version – for: United Airlines ﬂight 1377 visits

the following airports: LAX, DFW, and JFK.

10 Marco Giunti et al.

The Working Group Note maintains that the four examples diﬀer on two

diﬀerent respects. First, only in Example 4 would the order of the individuals

make a real diﬀerence. In the ﬁrst three examples, instead, individuals’ order

would not seem to be essential. Second, in Examples 1, 2, and 4 it is possible

to distinguish one of the individuals as the main subject of the whole relation

(respectively, Christine, Steve, and United Airlines ﬂight 1377), while in Exam-

ple 3 there is no “[...] single individual standing out as the subject or the ‘owner’

of the relation.” (W3C [23], Sect. 3.2.3).

Due to the ﬁrst diﬀerence, the Working Group Note proposes two diﬀerent

ontological patterns to represent, on the one hand, Examples 1, 2, 3, and on the

other one, Example 4. Both patterns are special versions of the relation reiﬁ-

cation model, which are expressed in either RDFS or OWL. The corresponding

representations of the four examples by means of the two ontological patterns

are shown in Figures 1, 2, 3, and 4.

The ﬁrst ontological pattern is nothing more than a straightforward RDFS or

OWL translation of the relation reiﬁcation model. This is evident from Figure 3,

which shows exactly the same kind of graph as the one employed by Deliyanni

and Kowalski for representing the statement “John gives the book to Mary”

([4], p. 184). The special relation-class Purchase (not represented in Figure 3)

is the reiﬁcation of the 5-ary relation ()buys a()from()for()as a(). The instance

Purchase_1 of the relation-class can be thought as the 5-tuple for which the

relation holds, and it is connected to each of the ﬁve individuals by an outgoing

arrow. The arrows stand for ﬁve binary relations whose respective meanings, as

well as the meaning of the relation-class Purchase, are speciﬁed in the RDFS

or OWL ontology, in such a way that they should collectively approximate the

meaning of the original 5-ary relation ()buys a()from()for()as a().

A slightly diﬀerent version of the ﬁrst pattern is provided for Examples 1

and 2, as shown in Figures 1 and 2. Recall that, according to the Working

Group Note, in either example one of the individuals would stand out as the

subject of the whole relation. This is indicated by introducing exactly one arrow

whose direction is reversed: the individual from which the reversed arrow orig-

inates is the subject, while all other arrows originate from the instance of the

relation-class. Also note that, like in Figure 3, the relation-class and the binary

relation (rdf:type) that connects its instance to it are not represented in either

Figure 1 or 2.

The second pattern is a somewhat more complex version of the relation-

reiﬁcation model, which takes care of the special feature of Example 4: “[...] all

but one participant in a relation do not have a speciﬁc role and essentially form

an ordered list” (W3C [23], Sect. 3.3).

The relation-class FlightSegment (not represented in Figure 4) is introduced

as the reiﬁcation of the original 4-ary relation ()visits the following airports:(),(),

and(). However, in order to take care of the special feature of Example 4, any

instance of the relation-class should now be thought as an ordered list of ﬂight

destinations. This leaves out United Airlines ﬂight 1377, the single participant

with a speciﬁc role, which does not belong to any destination list. United Air-

Representing n-ary relations 11

lines ﬂight 1377 can thus be thought as the subject of the whole relation, similar

to Christine and Steve in, respectively, Examples 1 and 2. In order to represent

the original n-ary fact, as many instances of the relation-class should be created

as the number of ordered individuals; thus, in this speciﬁc case, the three in-

stances UA_1377_1, UA_1377_2, UA_1377_3 are introduced. Each of them

should be thought as, respectively, the complete ﬂight destination list, the com-

plete list except its ﬁrst element, and the ﬁnal list whose only element is the

last destination. This is obtained by means of the two binary relations destina-

tion and next_segment, and the subclass FinalFlightSegment (not represented

in Figure 4) of the relation-class FlightSegment. The three destinations and the

corresponding instances of the relation-class are in fact connected to each other

in a typical list structure, as shown in Figure 4. The subclass FinalFlightSeg-

ment identiﬁes the ﬁnal destination list, for the third instance UA_1377_3 is

also an instance of this class. United Airlines ﬂight 1377, the subject of the whole

relation, is ﬁnally connected to the ﬁrst instance of the list structure by means

of the binary relation ﬂight_sequence. The meanings of the three binary rela-

tions, destination, next_segment and ﬂight_sequence, as well as those of the

two classes FlightSegment and FinalFlightSegment, are speciﬁed in the RDFS

or OWL ontology, in such a way that they should collectively approximate the

meaning of the original 4-ary relation ()visits the following airports:(),(), and().

4 An abstract graph model for representing n-ary

relations

In this Section, we present an abstract model for representing n-ary relations as

directed labeled graphs.4We will then reconsider the four use case examples of

the previous Section and show how this model leads to improved representation.

We have seen in Sect. 2 that a n-ary (n≥1) predicate is usually formalized

by means of an uppercase letter, for instance “R”, together with a superscripted

“n” that indicates its arity: “Rn”. For our present purposes, however, it is more

convenient to indicate the arity nof a predicate by means of a corresponding

number of left and right parentheses “()” to the right of the uppercase letter.

Thus, according to this notation, “Rn” will be written “R()...()”, where there are

exactly ncopies of “()” to the right of “R”.

Consider now an arbitrary n-ary (n≥3) relation R()()...(), indicated by the

predicate “R()()...()”. Let us ﬁrst of all recall that, according to the deﬁnition

given in Sect. 2, the n-ary relation R()()...() is a class, intensionally understood,

of ordered n-tuples. Let (a1, a2, ... , an)be an arbitrary element of R()()...(); we

express this n-ary fact (n≥3) with the statement “R(a1)(a2)...(an)”.5Now, we

ask how it is possible, in general, to represent the n-ary fact R(a1)(a2)...(an)by

means of statements that only involve binary relations.

4A seminal version of this model can be found in Giunti [9].

5It should be noticed that the quoted text “R(a1)(a2)...(an)” indicates a statement

that expresses a n-ary fact, while the unquoted text R(a1)(a2)...(an)states the fact

itself.

12 Marco Giunti et al.

Given the relational fact R(a1)(a2)...(an), we ﬁrst consider the individuals

a3, ..., an, and we then use them as parameters to deﬁne the binary relation

R()()[a3]...[an]by means of the n-ary one R()()...(). Formally, this is obtained

by deﬁning6the new binary predicate “R()()[a3]...[an]” by means of the n-ary

predicate “R()()...()”, the n−2individual constants “a3”, . . . , “ an”, and two

individual variables “x” and “y”, as follows:

R(x)(y)[a3]...[an] := R(x)(y)(a3)...(an)

We then consider again the relational fact R(a1)(a2)...(an)and we repeat the

above procedure, now taking as parameters all the individuals except the second

and the third, then all the individuals except the third and the fourth, and so on,

until the last two individuals are excluded. This is formally obtained by means of

the following deﬁnitions of the new n−2binary predicates “R[a1]()()[a4]...[an]”,

“R[a1][a2]()()[a5]...[an]”, . . . , “ R[a1]...[an−2]()()”:

R[a1](x)(y)[a4]...[an] := R(a1)(x)(y)(a4)...(an)

R[a1][a2](x)(y)[a5]...[an] := R(a1)(a2)(x)(y)(a5)...(an)

. . . := . . .

. . . := . . .

. . . := . . .

R[a1]...[an−2](x)(y) := R(a1)...(an−2)(x)(y)

Note now that, for each of the n−1binary predicates, its deﬁnition entails

the corresponding equivalence below:

R(a1)(a2)[a3]...[an]↔R(a1)(a2)(a3)...(an)

R[a1](a2)(a3)[a4]...[an]↔R(a1)(a2)(a3)(a4)...(an)

R[a1][a2](a3)(a4)[a5]...[an]↔R(a1)(a2)(a3)(a4)(a5)...(an)

. . . ↔. . .

. . . ↔. . .

. . . ↔. . .

R[a1]...[an−2](an−1)(an)↔R(a1)...(an−2)(an−1)(an)

6As usual, each of the deﬁnitions below should be intended as the non-creative axiom

that is obtained by replacing the deﬁnition sign “:=” with the biconditional “ ↔” and

by then taking the universal closure of the resulting open formula α(x, y), in which

the only two free variables are “x” and “y”.

Representing n-ary relations 13

This means that the n-ary fact R(a1)(a2)...(an)can be expressed by any of

the statements on the left hand side of the above equivalences. It is important

to notice that each of these statements only involves a binary relation whose

meaning is just a parametric speciﬁcation of the meaning of the n-ary relation

R()()...(), for it is obtained from it by taking as parameters n−2individuals

involved in the original n-ary fact R(a1)(a2)...(an). This means that the equiv-

alences above are not just extensional, but they are in fact fully intensional.

Furthermore, given the previous n−1equivalences, the following one obviously

follows:

R(a1)(a2)...(an)↔R(a1)(a2)[a3]...[an]∧R[a1](a2)(a3)[a4]...[an]∧. . .

. . . ∧R[a1]...[an−2](an−1)(an)(1)

Formula (1) gives us a formal solution to our problem. In accordance to it,

the n-ary fact R(a1)(a2)...(an)can always be represented by the conjunction of

n−1chained statements in which only binary relations are involved. The n−1

statements are chained in the sense that (i) each of them involves a parametric

binary relation and (ii) these parametric binary relations can be ordered in

such a way that the parameters of the ﬁrst relation R()()[a3]...[an]are all the

ordered individuals a1, a2, ..., anexcept the ﬁrst and the second, the parameters

of the second relation R[a1]()()[a4]...[an]are all the ordered individuals except

the second and the third, and so on, until the last relation R[a1]...[an−2]()(),

whose parameters are all the ordered individuals except the last two.

This kind of chained representation of R(a1)(a2)...(an), where the order of

the parametric binary relations reproduces the order of the individuals in the n-

tuple (a1, a2, ..., an), is of special interest to us, for it is immediately translatable

into an oriented labeled graph. Figure 5 shows the graph that corresponds to

the conjunction on the right of the biconditional in Formula (1).

R()()[a_3]...[a_n]

a_1 a_3a_2 a_n-1 a_n

R[a_1]()()[a_4]...[a_n]

R[a_1]...[a_n-2]()()

Fig. 5. The oriented labeled graph that corresponds to the chained conjunction in the

right hand side of Formula (1).

4.1 Applying the abstract graph model to the W3C Working Group

examples

We show below how our abstract graph model can be applied to the four W3C

Working Group examples discussed in Sect. 3.

14 Marco Giunti et al.

Example 1. Christine has breast tumor with high probability

(Christine) has (breast tumor) with (high) probability

H()()() = ()has()with()probability;c=Christine;t=breast tumor;h=high.

Deﬁnition of H()()[h]:H(x)(y)[h] := H(x)(y)(h)

Deﬁnition of H[c]()():H[c](x)(y) := H(c)(x)(y)

Thus we get: H(c)(t)(h)↔H(c)(t)[h]∧H[c](t)(h)

Figure 6 shows the graph representation that corresponds to the chained

conjunction on the right of the biconditional above:

H()()[h] H[c]()()

c ht

Fig. 6. The abstract graph model for: Christine has breast tumor with high probability.

Example 2. Steve has temperature, which is high, but falling

(Steve) has temperature, which is (high), but (falling)

T()()() = ()has temperature, which is(), but();s=Steve;h=high;f=falling.

Deﬁnition of T()()[f]:T(x)(y)[f] := T(x)(y)(f)

Deﬁnition of T[s]()():T[s](x)(y) := T(s)(x)(y)

Thus we get: T(s)(h)(f)↔T(s)(h)[f]∧T[s](h)(f)

Figure 7 shows the graph representation that corresponds to the chained

conjunction on the right of the biconditional above:

T()()[f] T[s]()()

s fh

Fig. 7. The abstract graph model for: Steve has temperature, which is high, but falling.

Example 3. John buys a “Lenny the Lion” book from books.example.com

for $15 as a birthday gift

(John) buys a (“Lenny the Lion” book) from (books.example.com) for ($15) as

a (birthday gift)

B()()()() = ()buys a()from()for()as a();j=John;l=“Lenny the Lion” book;b=

books.example.com;p=$15;g=birthday gift.

Deﬁnition of B()()[b][p][g]:B(x)(y)[b][p][g] := B(x)(y)(b)(p)(g)

Deﬁnition of B[j]()()[p][g]:B[j](x)(y)[p][g] := B(j)(x)(y)(p)(g)

Representing n-ary relations 15

Deﬁnition of B[j][l]()()[g]:B[j][l](x)(y)[g] := B(j)(l)(x)(y)(g)

Deﬁnition of B[j][l][b]()():B[j][l][b](x)(y) := B(j)(l)(b)(x)(y)

Thus we get: B(j)(l)(b)(p)(g)↔B(j)(l)[b][p][g]∧B[j](l)(b)[p][g]∧

B[j][l](b)(p)[g]∧B[j][l][b](p)(g)

Figure 8 shows the graph representation that corresponds to the chained

conjunction on the right of the biconditional above:

B()()[b][p][g]

j bl pg

B[j]()()[p][g]

B[j][l]()()[g]

B[j][l][b]()()

Fig. 8. The abstract graph model for: John buys a “Lenny the Lion” book from

books.example.com for $15 as a birthday gift.

Example 4. United Airlines ﬂight 1377 visits the following airports:

LAX, DFW, and JFK

(United Airlines ﬂight 1377) visits the following airports: (LAX), (DFW), and

(JFK)

V()()()() = ()visits the following airports:(),(), and();u=United Airlines ﬂight

1377; l=LAX;d=DFW;j=JFK.

Deﬁnition of V()()[d][j]:V(x)(y)[d][j] := V(x)(y)(d)(j)

Deﬁnition of V[u]()()[j]:V[u](x)(y)[j] := V(u)(x)(y)(j)

Deﬁnition of V[u][l]()():V[u][l](x)(y) := V(u)(l)(x)(y)

Thus we get: V(u)(l)(d)(j)↔V(u)(l)[d][j]∧V[u](l)(d)[j]∧V[u][l](d)(j)

Figure 9 shows the graph representation that corresponds to the chained

conjunction on the right of the biconditional above:

V()()[d][j]

u dl j

V[u]()()[j]

V[u][l]()()

Fig. 9. The abstract graph model for: United Airlines ﬂight 1377 visits the following

airports: LAX, DFW and JFK.

4.2 The abstract graph model vs. the two ontological patterns

based on relation reiﬁcation

In this subsection, we compare how the abstract graph model and the two on-

tological patterns based on relation reiﬁcation (see Sect. 3) respectively fare on

16 Marco Giunti et al.

three issues: the order problem, the meaning problem, and the subject problem.

We maintain that all three problems are serious for the reiﬁcation patterns, but

they are easily handled by the abstract graph model.

The order problem

We have seen in Sect. 2 that order is a fundamental feature of relations in general.

This is to be intended in the precise sense that any n-ary (n≥1) relation holds,

or does not, for any nindividuals a1, ... , anin a given order. In other words, the

order in which the individuals are considered is an essential feature of any n-ary

fact or, to put it in a diﬀerent way, it does not even make sense to ask whether

a relation holds for nindividuals if they are not listed in some order. Thus,

independently of the nature of the relation involved, the order of the individuals

is always relevant to single out a relational fact.

Quite obviously, this is not to say that order is always relevant for the relation

itself. For example, when a binary relation is symmetric, if it holds for two

individuals in a given order, it holds as well for the same individuals in the

reversed order. But note that symmetry is a property of the relation, it is not a

property of any single fact in which the symmetric relation is involved.

Ontological patterns for n-ary relations, as well as the abstract graph model,

are primarily designed to faithfully represent n-ary facts, and we have just argued

that individuals’ order is an essential feature of any n-ary fact. Let us then ask

how well the two reiﬁcation patterns on the one hand, and the abstract graph

model on the other one, are able to represent this essential feature of n-ary facts.

As far as the ﬁrst reiﬁcation pattern is concerned, it is quite obvious that there

is no way to retrieve any particular order of the individuals from a representation

of a n-ary fact based on either version of this pattern (subject version or no

subject one). One might think that this is not a serious problem, for the ﬁrst

pattern is supposed to be applied only to those n-ary facts for which individuals’

order does not matter. But we have just seen that there are no such facts, so that

when a n-ary fact is represented by means of this pattern, an essential feature

of the fact itself is irremediably lost.

The second reiﬁcation pattern, instead, is able to correctly represent indi-

viduals’ order. However, we have seen that this is obtained by increasing the

technical complexity of the pattern, which now includes a full-blown list struc-

ture, and by resorting to a quite unnatural interpretation of the relation-class.

In the ﬁrst reiﬁcation pattern, the elements of the relation-class can be thought

to be just the n-tuples for which the original n-ary relation holds. This natural

interpretation is no longer possible for the relation-class of the second pattern,

for now the class also has as elements all k-tuples of length 1≤k < n that are

obtained from the previous n-tuples by repeatedly discarding their ﬁrst items.

The abstract graph model, by contrast, represents the order of the individuals

of a n-ary fact in the most straightforward and natural way, for the n-tuple for

which the n-ary relation holds can be immediately retrieved from the connected

and oriented graph that represents the whole fact.

Representing n-ary relations 17

The meaning problem

We now consider the following question: How faithful are the two reiﬁcation

patterns, or the abstract graph model, to the intended meaning of the n-ary

relation involved in a fact they represent?

We have seen in Sect. 3 that, for either reiﬁcation pattern, the answer to

this question entirely depends on how well the ontologically speciﬁed meanings

of the relation-class and the specially introduced binary relations are going to

collectively approximate the meaning of the original n-ary relation. However,

the main problem with this approach is that it is not at all clear how a good

approximation might ever be reached, for neither pattern provides any principled

way to correlate the meanings of the relation-class and binary relations to the

one of the original n-ary relation.

For the abstract graph model, instead, the answer entirely depends on how

the meanings of the n−1parametric binary relations are related to the meaning

of the original n-ary relation. As noted above (Sec. 4, par. 7), the meaning of

each binary relation is correlated to the meaning of the n-ary relation in a clear

and principled way, for it is in fact a parametric speciﬁcation of the meaning

of the n-ary relation, which is obtained by taking as parameters n−2of the

individuals involved in the n-ary fact to be represented. As previously remarked,

it thus follows that the abstract graph model that represents a n-ary fact can

always be identiﬁed with the conjunction of n−1chained statements, where each

of them is intensionally equivalent to the statement that expresses the original

n-ary fact.

The subject problem

As mentioned above (Sect. 3, par. 6), according to the Working Group Note, it is

sometimes possible to distinguish one of the individuals involved in a relational

fact as the main subject of the n-ary relation. This motivates the introduction

of two diﬀerent versions of the ﬁrst pattern, as well as the special role accorded

to one of the individuals in the second pattern.

However, this particular feature of either pattern is in fact in sharp contrast

with the very notion of a relational fact, in which none of the individuals involved

can be thought as the subject or the “owner” of the relation, because it is just the

n-th entity for which the relation holds. For the abstract graph model, instead,

the subject problem does not even arise, because each node of its linear oriented

graph just plays the role of the n-th individual for which the relation holds.

5 PROL and the parametric pattern

We have seen in the previous section how the abstract graph model leads to im-

proved representation of a n-ary fact as regards at least three scores: (i) capacity

to represent individuals’ order in a simple and eﬀective way; (ii) faithfulness to

the meaning of the original n-ary relation; (iii) overall technical simplicity and

naturalness of the graph representation.

18 Marco Giunti et al.

In this ﬁnal section, we show how the abstract graph model gives rise to a

new ontological pattern (parametric pattern) for representing n-ary relations in

the Semantic Web. To this end, we preliminarily deﬁne PROL (Parametric Rela-

tional Ontology Language). PROL is a simple ontological language, compatible

with RDF, which is designed to express an arbitrary n-ary fact (n≥1) as a para-

metric pattern. The vocabulary of PROL includes just 6 terms (2 RDF classes

and 4 RDF properties) deﬁned by a simple RDFS ontology. Two terms (prol:type,

prol:next) serve to represent any n-ary fact as a parametric pattern. The re-

maining four terms (prol:Relation, prol:Domain, prol:hasPlaces, prol:represents)

serve to express the ontology that (a) deﬁnes the n-ary relations involved in the

facts to be represented, as well as the corresponding parametric binary relations,

and (b) allows for the correct detection and interpretation of the representing

parametric patterns.

5.1 PROL – Parametric Relational Ontology Language

In this subsection, we brieﬂy review the six terms of PROL and the RDF triples

which ﬁx their meaning. The complete RDFS ontology (TURTLE version) that

deﬁnes the PROL vocabulary is available at <http://semrapcon.altervista.org/

prol/prol.ttl>.7

The intended meaning of the term “prol:Relation” is the class of all n-ary

relations (n≥1). Recall that, according to the deﬁnition given in Sect. 2, any

n-ary relation is a class, intensionally understood, of ordered n-tuples. This can

be expressed by the following three RDF triples:

prol:Relation rdf:type rdfs:Class.

prol:Relation rdfs:subClassOf rdfs:Class.

prol:Relation rdfs:subClassOf rdfs:Seq.

The ﬁrst and the second triple state that prol:Relation is a RDF class whose

elements are RDF classes as well. The third one states that every n-ary relation

is also a RDF sequence-container, that is to say, an entity for which it makes

sense to specify a ﬁrst item, a second item, . . . , a n-th item, by means of the

RDF properties rdf:_1, rdf:_2, . . . , rdf:_n. Moreover, it is understood that any

such i-th item (1 ≤i≤n)is an i-th domain of the n-ary relation.

The intended meaning of the term “prol:Domain” is the class of all classes

that are domains of some n-ary relation. This is expressed by the following two

triples:

prol:Domain rdf:type rdfs:Class.

prol:Domain rdfs:subClassOf rdfs:Class.

The intended meaning of the term “prol:hasPlaces” is a RDF property that

expresses the arity nof a relation. Therefore, the domain of this property is the

7For the RDF/XML version: <http://semrapcon.altervista.org/prol/prol>.

Representing n-ary relations 19

class prol:Relation, while its range is the integer data-type of XML Schema, i.e.,

xsd:integer. This is conveyed by the following three triples:

prol:hasPlaces rdf:type rdf:Property.

prol:hasPlaces rdfs:domain prol:Relation.

prol:hasPlaces rdfs:range xsd:integer.

The intended meaning of the term “prol:represents” is a RDF property that

applies to a property and whose value is a n-ary relation of arity n≥2. The prop-

erty to which prol:represents applies is to be thought as the parametric binary

relation obtained from the n-ary relation by taking as parameters nindividuals

in a given order, except the ﬁrst and the second one. The n−2parameters will

be speciﬁed by means of the RDF properties rdf:_3, . . . , rdf:_n. When n= 2,

this property has no parameter, and it can thus be identiﬁed with the n-ary

relation itself. This is partly expressed by the following four triples:

prol:represents rdf:type rdf:Property.

prol:represents rdfs:domain rdf:Property.

prol:represents rdfs:domain rdfs:Seq.

prol:represents rdfs:range prol:Relation.

The ﬁrst three triples state that prol:represents is a RDF property, which

always applies to properties that are sequence-containers as well; the fourth one

states that the range of prol:represents is the class of all n-ary relations.

The term “prol:type” stands for a RDF property that applies to an individ-

ual, and has as value a n-ary (n≥1) relation. This is expressed by the following

three triples:

prol:type rdf:type rdf:property.

prol:type rdfs:domain rdfs:Resource.

prol:type rdfs:range prol:Relation.

It is intended that the individual to which prol:type applies is always the

ﬁrst node of an oriented linear graph, called parametric path, which is a straight-

forward RDF translation of an abstract graph model. The parametric path,

together with the triple that connects its ﬁrst node to the n-ary relation by

means of prol:type, form a parametric pattern, whose overall intended meaning

is that the n-ary relation holds for the n-tuple of individuals determined by the

parametric path.

The ﬁrst triple of a parametric path with n≥3individuals always has as

predicate the parametric property obtained from the n-ary relation by taking as

parameters all the nindividuals of the path except the ﬁrst and the second one.

All the remaining n−2triples of the parametric path have as predicate the term

“prol:next”.

20 Marco Giunti et al.

The intended meaning of the term “prol:next” is thus a RDF property that

applies to arbitrary individuals and whose values are arbitrary individuals as

well. This is expressed by the following three triples:

prol:next rdf:type rdf:Property.

prol:next rdfs:domain rdfs:Resource.

prol:next rdfs:range rdfs:Resource.

When a parametric path has just two individuals, it is formed by just one

triple, whose predicate is the parametric property with 0parameters obtained

from the binary relation by taking as parameters the 2individuals of the path

except the ﬁrst and the second one. This 0-parameter property can be identiﬁed

with the binary relation itself.

Finally, when a parametric path has just one individual, it does not include

any triple, and the whole parametric pattern thus reduces to the single triple

that connects the unique individual of the path to the unary relation by means

of the prol:type property.

5.2 How to deﬁne in PROL a n-ary relation, the corresponding

parametric property, and the parametric pattern that

represents a n-ary fact in which the relation is involved

Having deﬁned in RDFS all the terms of PROL, it is now possible to use them

to deﬁne n-ary relations, as well as the parametric properties that represent

them. Finally, we will be able to use these relations and properties, together

with the properties prol:type and prol:next, to implement parametric patterns

that represent corresponding n-ary facts. Depending on the arity n≥1of the

relation to be deﬁned, three diﬀerent cases must be considered. We give below

a typical example for each of the three cases. The complete ontology (TURTLE

version) with the three examples of this section, as well as those of Sect. 4, is

available at <http://semrapcon.altervista.org/prol/onto/prol_example.ttl>.8

Case 1 (n≥3). Example: Christine has breast tumor with high prob-

ability

Suppose we want to represent the statement: Christine has breast tumor with

high probability. By the analysis previously discussed (Sect. 3, par. 2) we get:

(Christine) has (breast tumor) with (high) probability. Thus, according to this

analysis, the given statement involves the three individuals Christine, breast

tumor, high, in this order, and the ternary relation ()has()with()probability.

The ternary relation ()has()with()probability can be deﬁned by means of

the following RDF triples, which employ the three terms prol:Relation,

prol:hasPlaces, prol:Domain, as well as terms of the RDF or RDFS vocabulary:

8For the RDF/XML version: <http://semrapcon.altervista.org/prol/onto/prol_

example>.

Representing n-ary relations 21

ex:R_-has_-with_-probability rdf:type prol:Relation.

ex:R_-has_-with_-probability rdfs:label "()has()with()probability"@en.

ex:R_-has_-with_-probability prol:hasPlaces "3"^^xsd:integer.

ex:R_-has_-with_-probability rdf:_1 ex:HumanBeing.

ex:HumanBeing rdf:type prol:Domain.

ex:HumanBeing rdfs:comment "The class of all human beings."@en.

ex:R_-has_-with_-probability rdf:_2 ex:Disease.

ex:Disease rdf:type prol:Domain.

ex:Disease rdfs:comment "The class of all diseases of human beings."@en.

ex:R_-has_-with_-probability rdf:_3 ex:QualProbValues.

ex:QualProbValues rdf:type prol:Domain.

ex:QualProbValues rdfs:comment "The class of all qualitative probability val-

ues."@en.

The ﬁrst two triples state that ex:R_-has_-with_-probability is a n-ary re-

lation whose English label is “()has()with()probability”, while the third triple

states that its arity is 3. The second three triples state that the ﬁrst domain

of the ternary relation ex:R_-has_-with_-probability is the class of all human

beings. The third three triples state that the second domain of the ternary

relation ex:R_-has_-with_-probability is the class of all diseases of human be-

ings. The last three triples state that the third domain of the ternary relation

ex:R_-has_-with_-probability is the class of all qualitative probability values.

We then deﬁne the parametric property which is obtained from the ternary

relation ()has()with()probability by taking as parameters the three individuals

involved in the fact to be represented, except the ﬁrst two. Thus, in this particular

case, there is just one parameter, the third individual, which is the qualitative

probability value high. The deﬁnition is expressed by the following two RDF

triples, which employ the term prol:represents and the RDF term rdf:_3:

ex:_-has_-with_highPr-probability

prol:represents ex:R_-has_-with_-probability.

ex:_-has_-with_highPr-probability rdf:_3 ex:highPr.

We are ﬁnally in the position to represent the ternary fact (Christine) has

(breast tumor) with (high) probability by means of an appropriate paramet-

ric pattern, whose triples employ the previously deﬁned ternary re-

lation ex:R_-has_-with_-probability and the parametric property

ex:_-has_-with_highPr-probability, as well as the two terms prol:type

and prol:next:

ex:christine prol:type ex:R_-has_-with_-probability.

ex:christine ex:_-has_-with_highPr-probability ex:breastTumor.

ex:breastTumor prol:next ex:highPr.

22 Marco Giunti et al.

The last two triples of the parametric pattern form the parametric path that

is a straightforward PROL translation of the abstract graph model of Figure 6.

The ﬁrst triple completes the parametric pattern by connecting the ﬁrst individ-

ual (Christine) of the parametric path to the ternary relation by means of the

property prol:type. The whole parametric pattern is shown in Figure 10.

ex:_-has_-with_highPr-probability

prol:type

prol:next

ex:cristine

ex:R_-has_-with_-probability

ex:highPrex:breastTumor

Fig. 10. The parametric pattern for: Christine has breast tumor with high probability

Case 2 (n= 2). Example: The Divine Comedy is written by Dante

Suppose we want to represent the statement: The Divine Comedy is written by

Dante. We ﬁrst analyze this statement in the obvious way: (The Divine Comedy)

is written by (Dante). Thus, according to this analysis, the given statement

involves the two individuals The Divine Comedy and Dante, in this order, and

the binary relation ()is written by().

The binary relation ()is written by() can be deﬁned by means of the fol-

lowing RDF triples, which employ the three terms prol:Relation, prol:hasPlaces,

prol:Domain, as well as terms of the RDF or RDFS vocabulary:

ex:R_-isWrittenBy_- rdf:type prol:Relation.

ex:R_-isWrittenBy_- rdfs:label "()is written by()"@en.

ex:R_-isWrittenBy_- prol:hasPlaces "2"^^xsd:integer.

ex:R_-isWrittenBy_- rdf:_1 ex:WrittenText.

ex:WrittenText rdf:type prol:Domain.

ex:WrittenText rdfs:comment "The class of all written texts."@en.

ex:R_-isWrittenBy_- rdf:_2 ex:WrittenTextAuthor.

ex:WrittenTextAuthor rdf:type prol:Domain.

ex:WrittenTextAuthor rdfs:comment "The class of all authors of written

texts."@en.

The ﬁrst two triples state that ex:R_-isWrittenBy_- is a n-ary relation whose

English label is “()is written by()”, while the third triple states that its arity is

2. The subsequent three triples state that the ﬁrst domain of the binary relation

Representing n-ary relations 23

ex:R_-isWrittenBy_- is the class of all written texts. The last three triples state

that the second domain of the binary relation ex:R_-isWrittenBy_- is the class

of all authors of written texts.

We then deﬁne the parametric property which is obtained from the binary

relation ()is written by() by taking as parameters the two individuals involved

in the fact to be represented, except the ﬁrst two. Thus, in this case, there is

no parameter. The deﬁnition is expressed by the following RDF triple, which

employs the term prol:represents:

ex:_-isWrittenBy_- prol:represents ex:R_-isWrittenBy_-.

We are ﬁnally in the position to represent the binary fact (The Divine Com-

edy) is written by (Dante) by means of an appropriate parametric pattern, whose

triples employ the previously deﬁned binary relation ex:R_-isWrittenBy_- and

the parametric property ex:_-isWrittenBy_-, as well as the term prol:type:

ex:divineComedy prol:type ex:R_-isWrittenBy_-.

ex:divineComedy ex:_-isWrittenBy_- ex:dante.

The parametric pattern for the case n= 2 is shown in Figure 11.

ex:_-isWrittenBy_-

prol:type

ex:divineComedy

ex:R_-isWrittenBy_-

ex:dante

Fig. 11. The parametric pattern for: The Divine Comedy is written by Dante

Case 3 (n= 1). Example: Pluto is a dog

Suppose we want to represent the statement: Pluto is a dog. We ﬁrst analyze this

statement in the obvious way: (Pluto) is a dog. Thus, according to this analysis,

the given statement involves the individual Pluto and the unary relation ()is a

dog.

The unary relation ()is a dog can be deﬁned by means of the following RDF

triples, which employ the three terms prol:Relation, prol:hasPlaces, prol:Domain,

as well as terms of the RDF or RDFS vocabulary:

ex:R_-dog rdf:type prol:Relation.

24 Marco Giunti et al.

ex:R_-dog rdfs:label "()is a dog"@en.

ex:R_-dog prol:hasPlaces "1"^^xsd:integer.

ex:R_-dog rdf:_1 ex:Animal.

ex:Animal rdf:type prol:Domain.

ex:Animal rdfs:comment "The class of all animals."@en.

The ﬁrst two triples state that ex:R_-dog is a n-ary relation whose english

label is “()is a dog”, while the third triple states that its arity is 1. The last three

triples state that the domain of the unary relation ex:R_-dog is the class of all

animals.

We are ﬁnally in the position to represent the unary fact (Pluto) is a dog by

means of an appropriate parametric pattern. In this case (n= 1), the parametric

path has just one individual, so that it does not include any triple. The whole

parametric pattern thus reduces to the single triple that connects the unique in-

dividual of the path to the unary relation ex:R_-dog by means of the prol:type

property:

ex:pluto prol:type ex:R_-dog.

The parametric pattern for the case n= 1 is shown in Figure 12.

prol:type

ex:pluto

ex:R_-dog_-

Fig. 12. The parametric pattern for: Pluto is a dog

6 Conclusion

The problem of representing n-ary relations in the Semantic Web originates

from a fundamental expressive limit of RDF based ontological languages, such

as RDFS and OWL, which only allow for the direct representation of binary

relations. In this paper, we have developed a new twofold solution to this prob-

lem. First (sec. 4), we have described an abstract graph model for representing

n-ary relations. Through this model, it is possible to represent an arbitrary n-ary

fact, in a standard and principled way, by means of statements that only involve

Representing n-ary relations 25

binary relations. Furthermore, our model fares better than the usual relation

reiﬁcation model on three issues: the order problem, the meaning problem and

the subject problem. Second (sec. 5), we have shown how our abstract graph

model can be transformed into an actual ontological pattern, the parametric

pattern, by means of PROL, a new and simple ontological language compatible

with RDF.

As a ﬁnal remark, we would like to add that the parametric pattern can

also be used to express all the standard connectives and quantiﬁers of a ﬁrst

order language. This can be obtained by means of FOOL (First Order Ontology

Language), a natural ontological extension of PROL, compatible with RDF and

at least as expressive as any ﬁrst order language. Each ﬁrst order formula, once

expressed in FOOL, turns out to be a connected RDF graph made up of the

parametric patterns that represent all its subformulas (Giunti [10]; Giunti et al.

[11]). These further developments lead us to conjecture that it will eventually

be possible to express the whole complex of our knowledge as linked data, thus

actually realizing the original vision of the Semantic Web.

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