=Paper=
{{Paper
|id=Vol-1459/paper19
|storemode=property
|title=Towards fuzzy granulation in OWL ontologies
|pdfUrl=https://ceur-ws.org/Vol-1459/paper19.pdf
|volume=Vol-1459
|dblpUrl=https://dblp.org/rec/conf/cilc/LisiM15
}}
==Towards fuzzy granulation in OWL ontologies==
https://ceur-ws.org/Vol-1459/paper19.pdf
Towards Fuzzy Granulation in OWL Ontologies
Francesca A. Lisi and Corrado Mencar
Dipartimento di Informatica, Università degli Studi di Bari “Aldo Moro”, Italy
{francesca.lisi,corrado.mencar}@uniba.it
Abstract. The integration of fuzzy sets in ontologies for the Semantic
Web can be achieved in different ways. In most cases, fuzzy sets are
defined by hand or with some heuristic procedure that does not take into
account the distribution of available data. In this paper, we describe a
method for introducing a granular view of data within an OWL ontology.
1 Introduction
Endowing OWL ontologies with capabilities of representing and processing im-
precise knowledge is a highly desirable feature since the Semantic Web is full
of imprecise and uncertain information coming from perceptual data (i.e., data
coming from subjective judgments), incomplete data, data with errors, etc. [16].
Moreover, even in the case that precise information is available, imprecise knowl-
edge could be advantageous: tolerance to imprecision may lead to concrete bene-
fits such as compact knowledge representation, and efficient and robust reasoning
[20]. Additionally, humans continually acquire, manipulate and communicate im-
precise knowledge: therefore any ontology capable of expressing imprecise knowl-
edge, when a precise alternative leads to a complex representation, could be more
interpretable by human users, i.e. easier to read and understand [1].
A number of mathematical tools are available to deal with imprecision and
uncertainty in knowledge representation. The choice of the right tool depends on
the type of imprecision. In particular, imprecision due to the lack of boundaries in
concepts (such as coldness in the domain of indoor temperatures, interestingness
of movies, etc.) are well modeled through fuzzy sets. In essence, fuzzy sets define
collections of objects whose membership can be partial. Differently to probability
measures, the degree of membership does not measure how likely an object is
referred by a concept, but rather it quantifies how much the concept is applicable
to the object. Fuzziness pervades human reasoning and allows it to intelligently
act in complex environments: since fuzzy sets make possible a computational
representation of concepts with no sharp boundaries, they enable machines to
carry out human-centered information processing and reasoning [5].
The integration of fuzzy sets in ontologies for the Semantic Web can be
achieved in different ways (see [19] for an updated overview). However, in most
cases, fuzzy sets are defined by hand or with some heuristic procedure that does
not take into account the distribution of available data. In this paper, we propose
the adoption of a fuzzy clustering procedure to automatically acquire fuzzy sets
from data. Also, we exploit the resulting clusters, together with fuzzy quantifiers,
to develop a granular view of the individuals in an OWL ontology.
The paper is structured as follows. Section 2 presents some preliminary in-
formation on Description Logics1 (2.1), Fuzzy Set Theory (2.2), and Fuzzy DLs
(2.3). Section 3 describes the proposed granulation method on OWL schemas at
increasing levels of complexity. Finally, Section 4 draws some conclusive remarks
along with future research directions.
2 Preliminaries
2.1 Description Logics
Description Logics (DLs) are a family of decidable First Order Logic (FOL)
fragments that allow for the specification of structured knowledge in terms
of classes (concepts), instances (individuals), and binary relations between in-
stances (roles) [2]. Complex concepts (denoted with C) can be defined from
atomic concepts (A) and roles (R) by means of the constructors available for
the DL in hand. The members of the DL family differ from each other as for
the set of constructors, thus for the complexity of concept expressions they can
generate. For the sake of illustrative purposes, we present here a salient represen-
tative of the DL family, namely ALC [15], which is often considered to illustrate
some new notions related to DLs. A DL Knowledge Base (KB) K = hT , Ai is a
pair where T is the so-called Terminological Box (TBox) and A is the so-called
Assertional Box (ABox). The TBox is a finite set of General Concept Inclusion
(GCI) axioms which represent is-a relations between concepts, whereas the ABox
is a finite set of assertions (or facts) that represent instance-of relations between
individuals (resp. couples of individuals) and concepts (resp. roles). Thus, when
a DL-based ontology language is adopted, an ontology is nothing else than a
TBox (i.e., the intensional level of knowledge), and a populated ontology corre-
sponds to a whole KB (i.e., encompassing also an ABox, that is, the extensional
level of knowledge).
The semantics of DLs can be defined directly with set-theoretic formalizations
or through a mapping to FOL (as shown in [8]). Specifically, an interpretation
I = (∆I , ·I ) for a DL KB consists of a domain ∆I and a mapping function
·I . For instance, I maps a concept C into a set of individuals C I ⊆ ∆I , i.e. I
maps C into a function C I : ∆I → {0, 1} (either an individual belongs to the
extension of C or does not belong to it). Under the Unique Names Assumption
(UNA) [13], individuals are mapped to elements of ∆I such that aI 6= bI if
a 6= b. However UNA does not hold by default in DLs. An interpretation I is a
model of a KB K iff it satisfies all axioms and assertions in T and A. In DLs a
KB represents many different interpretations, i.e. all its models. This is coherent
with the Open World Assumption (OWA) that holds in FOL semantics. A DL
KB is satisfiable if it has at least one model. We also write C vK D if in any
1
We recap that DLs are the logical foundation of the standard for web ontology
languages belonging to the OWL 2 family [12].
Table 1. Syntax and semantics of constructs for ALC(D).
bottom (resp. top) concept ⊥ (resp. >) ∅ (resp. ∆I )
atomic concept A A I ⊆ ∆I
abstract role R R I ⊆ ∆I × ∆I
concrete role T T I ⊆ ∆I × ∆D
individual a aI ∈ ∆I
concrete value v v I ∈ ∆D
concept intersection CuD C I ∩ DI
concept union CtD C I ∪ DI
concept negation ¬C ∆I \ C I
universal abstract role restriction ∀R.C {x ∈ ∆I | ∀y (x, y) ∈ RI → y ∈ C I }
existential abstract role restriction ∃R.C {x ∈ ∆I | ∃y (x, y) ∈ RI ∧ y ∈ C I }
universal concrete role restriction ∀T.d {x ∈ ∆I | ∀z (x, z) ∈ T I → z ∈ dD }
existential concrete role restriction ∃T.d {x ∈ ∆I | ∃z (x, z) ∈ T I ∧ z ∈ dD }
general concept inclusion CvD C I ⊆ DI
concept assertion a:C aI ∈ C I
abstract role assertion (a, b) : R (aI , bI ) ∈ RI
concrete role assertion (a, v) : T (aI , v I ) ∈ T I
model I of K, C I ⊆ DI (concept C is subsumed by concept D). Moreover we
write C @K D if C vK D and D 6vK C. The consistency check, which tries to
prove the satisfiability of a DL KB K, is the main reasoning task in DLs. It is
performed by applying decision procedures mostly based on tableau calculus.
All other reasoning tasks can be reformulated as consistency checks.
In many applications, it is important to equip DLs with expressive means that
allow to describe “concrete qualities” of real-world objects such as the length of a
car. The standard approach is to augment DLs with a so-called concrete domain
(or datatype theory) D = h∆D , · D i, which consists of a datatype domain ∆D
(e.g., the set of real numbers in double precision) and a mapping · D that assigns
to each data value an element of ∆D , and to every n-ary datatype predicate d
an n-ary (typically, n = 1) relation over ∆D [3]. In DLs extended with concrete
domains, each role is therefore either abstract (denoted with R) or concrete
(denoted with T ). The set of constructors for ALC(D) is reported in Table 1.
2.2 Fuzzy Set Theory
A crisp set A over a Universe of Discourse X is characterised by a function
A : X → {0, 1}, that is, for any x ∈ X either x ∈ A (i.e., A(x) = 1) or x 6∈ A
(i.e., A(x) = 0). A fuzzy set F over X is characterised by a membership function
F : X → [0, 1]. For a fuzzy set F , unlike crisp sets, x ∈ X belongs to F to a
degree F (x) in [0, 1].
The trapezoidal, the triangular, the left-shoulder, and the right-shoulder
functions are frequently used as membership functions of fuzzy sets (see Fig. 1
for a graphical representation). In particular, the trapezoidal function is defined
1 1 1 1
0 0 0 0
a b c d x a b c x a b x a b x
(a) (b) (c) (d)
Fig. 1. Four notable membership functions of fuzzy sets: (a) Trapezoidal, (b) triangu-
lar, (c) left-shoulder, and (d) right-shoulder.
as follows: let a < b ≤ c < d be rational numbers then
0 if x < a
(x − a)/(b − a) if x ∈ [a, b)
trz(a, b, c, d)(x) = 1 if x ∈ [b, c] (1)
(d − x)/(d − c) if x ∈ (c, d]
0 if x > d .
Note that triangular, left-shoulder and right-shoulder fuzzy sets are special cases
of trapezoidal fuzzy sets.2
Fuzzy sets can be used to represent information granules, i.e. collections
of objects kept together due to their similarity, proximity, etc. [4]. Information
granules promote abstraction as far as they can be labeled by symbolic terms.
Information granules represented by fuzzy sets are good candidates to represent
perceptual information, thus they could be conveniently labeled by linguistic
terms coming from natural language. The granularity level (quantifiable as the
area of the membership function, for fuzzy sets) assesses the specificity of an
information granule: the most specific information granule is a set with a single
element (precise information); on the other extreme, an information granule
covering the whole universe of discourse is the least specific.
Fuzzy clustering Although fuzzy sets have a greater expressive power than
crisp sets, their usefulness depends critically on the capability to construct ap-
propriate membership functions for various given concepts in different contexts.
The problem of constructing meaningful membership functions is not a trivial
one (see, e.g., [10, Chapter 10]). One easy method is to define a uniform Strong
Fuzzy Partition (SFP) usually with 5 ± 2 fuzzy sets. A SFP is a collection of
fuzzy sets (usually with triangular or trapezoidal fuzzy sets) such that, for each
element of the Universe of Discourse, the sum of memberships of all fuzzy sets
is always 1. SFPs with trapezoidal fuzzy sets greatly enhance the efficiency of
calculations because they guarantee that each element has non-zero membership
degree for at most two fuzzy sets. A uniform SFP is based on fuzzy sets with the
same granularity. It is very simple to define a uniform SFP, but this approach
does not take into account the distribution of available data; in fact, coarse
2
By convention, whenever the denominator of one of the fractions in (1) is 0, the
membership degree is 1.
grained fuzzy sets are more useful to cover regions of the Universe of Discourse
where data are more sparse; on the other hand, data crammed in small areas
are better represented by more specific (fine grained) fuzzy sets.
The derivation of a SFP with variable granularity, adapted to available data,
can be achieved through fuzzy clustering. A widespread algorithm for fuzzy clus-
tering is Fuzzy C-Means (FCM) [6], an extension of the well-known K-Means
that accommodates partial memberships of data to clusters. FCM, applied to
one-dimensional, numerical data, can be used to derive a set of c clusters char-
acterized by prototypes π1 , π2 , . . . , πc , with πj ∈ R and πj < πj+1 . These proto-
types, along with the range of data, provide enough information to define a SFP
with two trapezoidal fuzzy sets and c − 2 triangular fuzzy sets according to the
following rules:
trz (m, m, π1 , π2 ) if j = 1
Fj = trz (πj−1 , πj , πj , πj+1 ) if 1 < j < c (2)
trz (πc−1 , πc , M, M ) if j = c.
where [m, M ] is the range of data. In Fig. 2 an example of SFP, consisting of
five fuzzy sets with variable granularity, is depicted.
Fig. 2. Example of SFP consisting of c = 5 fuzzy sets with variable granularity.
Fuzzy quantifiers Fuzzy sets, like crisp sets, can be quantified in terms of
their cardinality. Several definitions of cardinality of fuzzy sets are possible [9],
although in this paper we consider only relative scalar cardinalities like the
relative σ-count, defined for a finite Universe of Discourse X as:
P
F (x)
σ(F ) = x∈X (3)
|X|
A relative scalar cardinality yields a value within [0, 1] (being |∅| = 0 and
|X| = 1). On this interval, a number of fuzzy sets can be defined to repre-
sent granular concepts about cardinalities, such as Many (see Fig. 3 for some
notable examples). These concepts are called fuzzy quantifiers. Note that the
usual existential quantifier (∃) and universal quantifier (∀) can be represented
as special cases of fuzzy quantifiers: Q∃ (x) = 1 iff x > 0, 0 otherwise; Q∀ (x) = 1
iff x = 1, 0 otherwise.
Fig. 3. Some notable fuzzy quantifiers: None, Few, Some, Many, and Most.
Given a fuzzy quantifier Q and a fuzzy set F , the membership degree Q(σ(F ))
can be intended as the degree of truth of a fuzzy proposition of the form “Qx
are F ” (e.g.“Many x are Tall”).
2.3 Fuzzy Description Logics
In fuzzy DLs, an interpretation I = (∆I , ·I ) consists of a nonempty (crisp) set
∆I (the domain) and of a fuzzy interpretation function ·I that, e.g., maps a
concept C into a function C I : ∆I → [0, 1] and, thus, an individual belongs to
the extension of C to some degree in [0, 1], i.e. C I is a fuzzy set. The definition
of ·I for ALC(D) with fuzzy concrete domains is reported in [18]. In particular,
· D maps each concrete role into a function from ∆D to [0, 1]. Typical examples
of datatype predicates are
d := ls(a, b) | rs(a, b) | tri(a, b, c) | trz(a, b, c, d) | ≥v | ≤v | =v , (4)
where e.g. ≥v corresponds to the crisp set of data values that are greater or
equal than the value v.
Axioms in a fuzzy ALC(D) KB K = hT , Ai are graded, e.g. a GCI is of
the form hC1 v C2 , αi (i.e. C1 is a sub-concept of C2 to degree at least α). We
may omit the truth degree α of an axiom; in this case α = 1 is assumed. An
I
interpretation I satisfies an axiom hτ, αi if (τ ) ≥ α. I is a model of K iff
I satisfies each axiom in K. We say that K entails an axiom hτ, αi, denoted
K |= hτ, αi, if any model of K satisfies hτ, αi. Further details of the reasoning
procedures for fuzzy DLs can be found in [17].
Fuzzy quantifiers have been also studied in fuzzy DLs. In particular, Sanchez
and Tettamanzi [14] define an extension of fuzzy ALC(D) involving fuzzy quan-
tifiers of the absolute and relative kind, and using qualifiers. They also provide
algorithms for performing two important reasoning tasks with their DL: Reason-
ing about instances, and calculating the fuzzy satisfiability of a fuzzy concept.
3 Fuzzy Granulation of OWL Schemas
In this Section we show our proposal of introducing a granular view within an
ontology. We shall proceed incrementally starting from the simplest case. For
the sake of simplicity, we shall use the OWL terminology henceforth instead of
the DL terminology (We remind the reader that class stands for concept, and
property stands for role).
3.1 Case 1
Let C be a class and T a functional datatype property connecting instances of
C to values in a numerical range d. See Fig. 4 for a graphical representation of
this costruct.
Fig. 4. Graphical representation of a functional datatype property T with domain C
and range over a numerical datatype d.
This schema can be directly translated into a table (see Table 2) with two
columns and as many rows as the number of individuals of C for which T holds.
Table 2. Tabular representation of the OWL schema depicted in Fig. 4.
C T
a1 v1
a2 v2
··· ···
an vn
The dataset in Table 2 can be easily granulated in a number of fuzzy sets
F1 , F2 , . . . , Fc by applying, e.g., the fuzzy clustering method mentioned in Sec-
tion 2.2. In essence, the granulation process puts individuals in the same in-
formation granule if their respective values are similar. The use of fuzzy sets
to define granules ensures a gradual membership degree of individuals to such
granules, where the maximal membership is assigned to individuals detected as
“prototypes” of each granule. Each fuzzy set represents a fuzzy concept, and can
be tagged by a linguistic term, e.g. Low.
Table 3. Granulated individuals obtained from Table 2.
C F1 F2 · · · Fc
a1 µ11 µ12 · · · µ1c
a2 µ21 µ22 · · · µ2c
··· ··· ··· ··· ···
an µn1 µn2 · · · µnc
The result of granulation can be represented in a new table (see Table 3),
where each individual ai is associated to a row of membership values µij , being
µij = Fj (vi ) (5)
For each granule Fj , the relative cardinality σ(Fj ) can be computed by means
of the formula in Eq. (3). Given a fuzzy quantifier Qk , the membership degree
qjk = Qk (σ(Fj )) (6)
identifies the degree of truth of the fuzzy proposition “Qk x are Fj ”. In this
way, a new table can be constructed from a collection Q1 , Q2 , . . . , Qm of fuzzy
quantifiers, as shown in Table 4. If cm n, a sensible reduction of data can
be achieved to represent the original property through a granulated view. (To
further reduce data, a threshold τ can be set, so that all qjk less than τ are set
to zero.)
Table 4. Quantified cardinalities for the granules reported in Table 3.
Q1 Q2 · · · Qm
F1 q11 q12 · · · q1m
F2 q21 q22 · · · q2m
··· ··· ··· ··· ···
Fc qc1 qc2 · · · qcm
The new granulated view can be integrated in the ontology as follows. The
fuzzy sets Fj are the starting point for the definition of new subclasses of C
defined as Dj ≡ C u∃T.Fj . Also, a new class Granule is defined, with individuals
g1 , g2 , . . . , gc , where each individual gj is an information granule corresponding
to Fj . Each individual in Dj is then mapped to gj by means of an object property
mapsTo. Also, the cardinality of information granules is modeled by means of a
datatype property hasCardinality with domain in Granule and range in the
datatype domain xsd:double. Moreover, for each fuzzy quantifier Qk , a new
class is introduced, which models one of the fuzzy sets over the cardinalities
of information granules. The connection between the class Granule and each
class Qk is established through hasCardinality, once fuzzified, with degrees
identified as in Table 4. Note that the fuzzy proposition “Qk x are Fj ” is then
represented as the fuzzy assertion gj : ∃hasCardinality.Qk .
Example 1. In the tourism domain, we might consider an OWL ontology which
encompasses the datatype property hasPrice with the class Hotel as domain
and range in the datatype domain xsd:double. Let us suppose that the room
price for Hotel Verdi (instance verdi of Hotel) is 105, i.e. the KB contains the
assertion (verdi, 105):hasPrice. By applying fuzzy clustering to hasPrice,
we might obtain three fuzzy sets (Low, Medium, High) from which the following
classes are derived
LowPriceHotel ≡ Hotel u ∃hasPrice.Low
MidPriceHotel ≡ Hotel u ∃hasPrice.Medium
HighPriceHotel ≡ Hotel u ∃hasPrice.High.
With respect to these classes verdi shows different degrees of membership, e.g.
verdi is a low-price hotel at degree 0.8 and a mid-price hotel at degree 0.2 (see
Fig. 5 for a graphical representation). Subsequently, we might be interested in
obtaining aggregated information about hotels. Here the class Granule comes
into play. Quantified cardinalities allow us, for instance, to represent the fact that
“Many hotels are low-price” as the fuzzy assertion lph : ∃hasCardinality.Many
with truth degree 0.7, where lph is an instance of Granule (i.e., it is an infor-
mation granule) which corresponds to LowPriceHotel, and Many is one of the
fuzzy sets obtained from hasCardinality. Note that verdi, being an instance of
LowPriceHotel, maps to lph, i.e. (verdi, lph) : mapsTo holds to some degree.
3.2 Case 2
A natural extension of the proposed granulation method follows when the class
C is specialized in subclasses, as in Fig. 6. In this case, there are as many tables
with the same structure of Table 2 as the number of subclasses.
Analogously, for each subclass SubCj a structure of fuzzy information gran-
ules Fj1 , Fj2 , . . . , Fjc is produced and quantified according to the usual fuzzy
quantifiers Q1 , Q2 , . . . , Qm . (The quantifiers do not depend on the subclass as
their definition is fixed for all information granules.)
Example 2. Following Example 1, one may think of having a subsumption hierar-
chy with the class Accommodation as the root and Hotel and B&B as subclasses
(see Fig. 7). Hotels are granulated in three fuzzy subclasses (LowPriceHotel,
MidPriceHotel and HighPriceHotel) while B&Bs are granulated in two fuzzy
subclasses (CheapB&B and ExpensiveB&B). These fuzzy classes are related to the
classes representing fuzzy quantifiers via Granule analogously to Example 1.
3.3 Case 3
A case of particular interest is given by OWL schemas representing ternary re-
lations. A ternary relation is a subset of the Cartesian product involving three
domains C × D × N (for our purposes, we will assume N a numerical domain).
Fig. 5. Graphical representation of the output of the fuzzy granulation process on the
OWL schema described in Fig. 4 and instantiated with concepts reported in Example 1.
Fuzzy classes are depicted in gray.
Because of DL restrictions, however, ternary relations are not directly repre-
sentable in OWL, yet they can be indirectly represented through an auxiliary
class E, two object properties R1 and R2 , and one datatype property T , as
depicted in Fig. 8.
The structure in Fig. 8 corresponds to a tabular representation with three
columns, and as many rows as the number of elements of the relation, as in
table 5. By removing one of the two columns in Table 5, the resulting table is in
accordance with Table 2, which was the starting point of the granulation process.
In particular, as in the previous cases, a number of fuzzy sets F1 , F2 , . . . , Fc can
be derived starting from the dataset represented in Table 5, where one column
has been dropped. (We henceforth assume to drop column C.)
In order to connect information granules with classes, we proceed as follows.
Given an individual a ∈ C, a subset of Table 5 can be obtained, as in Table 6.
Fig. 6. Variant of the OWL schema shown in Fig. 4 for the case of C having subclasses.
Table 5. Tabular representation of the OWL schema depicted in Fig. 8.
C D T
a1 b1 v1
··· ··· ···
ai bj vk
··· ··· ···
an bm vl
More precisely, for each information granule Fj , it is possible to compute the
relative cardinality Pna
Fj (vai )
σja = i=1 (7)
na
Such cardinality can be quantified according to the fuzzy quantifiers Q1 , . . . , Qm .
The result is a table similar to Table 4, but now related to the individual a.
Table 6. A slice of Table 5 obtained by fixing an individual a in C.
C D T
a ba1 va1
a ba2 va2
··· ··· ···
a bana vana
Information granules, connected with the individuals in C, are arranged in
the ontology in a way that merges the modeling of ternary relations as in Fig. 8
with the granular model illustrated in case 1. The new classes, representing
information granules, are defined as Ej0 ≡ E u∃R2 .Du∃T.Fj . They are connected
to E in order to express a granular view of the relation between C and D.
Finally, a natural extension of this case allows the specialization of the class D
in subclasses (as in case 2).
Fig. 7. Graphical representation of the output of the fuzzy granulation process on the
OWL schema reported in Fig. 6 and instantiated with the concepts used in Example 2.
Example 3. With reference to the touristic domain, we might also consider the
distances between hotels and attractions (see Fig. 9). This is clearly a case
of ternary relation which requires to be modeled through an auxiliary class
Distance which is connected to the classes Hotel and Attraction by means of
the object properties hasDistance and isDistanceFor, respectively, and plays
the role of domain for a datatype property hasValue with range xsd:double.
The knowledge that “Hotel Verdi has a distance of 100 meters from the British
Museum” can be therefore modeled as follows:
(verdi, d1) : hasDistance
(d1, british museum) : isDistanceFor
(d1, 100) : hasValue
After fuzzy granulation, the imprecise sentence “Hotel Verdi has a low dis-
tance from many attractions” can be considered as a consequence of the previous
and the following axioms and assertions
LowDistance ≡ Distance u ∃isDistanceFor.Attraction u ∃hasValue.Low
d1 : LowDistance (to some degree)
(d1, ld) : mapsTo
(ld, 0.5) : hasCardinality
ld : ∃hasCardinality.Many (to some degree)
where Many is defined as mentioned in Example 1.
Fig. 8. Graphical representation of the OWL schema modeling a ternary relation.
4 Conclusions
This paper presents a starting point for introducing a granular view of data
within an OWL ontology. According to the ideas presented in the paper, a num-
ber of individuals belonging to the ontology can be replaced by information
granules, represented as fuzzy sets. In particular, the connection between the
existing individuals (not involved in granulation) and the granulated view is
possible by exploiting the peculiar representation of ternary relations in OWL.
This work is in a preliminary stage. We are currently evaluating the possibil-
ity of representing the output of our fuzzy granulation method by using OWL
2, i.e. within the current Semantic Web languages as suggested by Bobillo and
Straccia in their proposal of Fuzzy OWL 2 [7]. In a certain sense, we are pursuing
an alternative direction in comparison with the work of Sanchez and Tettamanzi
[14] which, if implemented, could lead to the extension of current Semantic Web
languages. However, it should be noted that there are some non-negligible restric-
tions to make their approach work in current fuzzy DLs. Notably, the approach
considers a finite number of individuals, which causes a mismatch with the usual
semantics for DLs (i.e., OWA with infinite interpretations).
Future research is aimed at integrating our fuzzy granulation approach within
inductive learning algorithms, such as Foil-DL [11], with the aim of verifying
the benefits of information granulation in terms of efficiency and effectiveness of
the learning process, as well as in terms of interpretability of the learning results.
Acknowledgements This work was partially funded by the Università degli
Studi di Bari “Aldo Moro” under the IDEA Giovani Ricercatori 2011 grant
“Dealing with Vague Knowledge in Ontology Refinement”.
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Fig. 9. Graphical representation of the output of the fuzzy granulation process on the
OWL schema reported in Fig. 8 and instantiated with the concepts used in Example 3.