A Fuzzy Description Logic for the Semantic Web
Umberto Straccia
ISTI-CNR
Via G. Moruzzi 1, I-56124 Pisa, ITALY
straccia@isti.cnr.it
Abstract
In this paper we present a fuzzy version of SHOIN (D), the corresponding
Description Logic of the ontology description language OWL
DL. We show that the representation and reasoning capabilities of fuzzy
SHOIN (D) go clearly beyond classical SHOIN (D). Interesting features
are: (i) concept constructors are based on t-norm, t-conorm, negation and
implication; (ii) concrete domains are fuzzy sets; (iii) fuzzy modifiers are
allowed; and (iv) entailment and subsumption relationships may hold to
some degree in the unit interval [0, 1].
Keywords: description logics, ontoloies, fuzzy logics
1 Introduction
In the last decade a substantial amount of work has been carried out in the
context of Description Logics (DLs) [1]. DLs are a logical reconstruction of
the so-called frame-based knowledge representation languages, with the aim of
providing a simple well-established Tarski-style declarative semantics to capture
the meaning of the most popular features of structured representation of
knowledge.
Nowadays, DLs have gained even more popularity due to their application in
the context of the Semantic Web [3, 16]. Semantic Web has recently attracted
much attention both from academia and industry, and is widely regarded as the
next step in the evolution of the World Wide Web. It aims at enhancing content
on the World Wide Web with meta-data, enabling agents (machines or human
users) to process, share and interpret Web content.
Ontologies [9] play a key role in the Semantic Web and major effort has been
put by the Semantic Web community into this issue. Informally, an ontology
consists of a hierarchical description of important concepts in a particular domain,
along with the description of the properties (of the instances) of each
concept. DLs play a particular role in this context as they are essentially the
theoretical counterpart of the Web Ontology Language OWL DL, the state of
the art language to specify ontologies. Web content is then annotated by relying
on the concepts defined in a specific domain ontology.
1
However, OWL DL becomes less suitable in all those domains in which the
concepts to be represented have not a precise definition. If we take into account
that we have to deal with Web content, then it is easily verified that this scenario
is, unfortunately, likely the rule rather than an exception. For instance, just
consider the case we would like to build an ontology about flowers. Then we
may encounter the problem of representing concepts like 1 “Candia is a creamy
white rose with dark pink edges to the petals”, “Jacaranda is a hot pink rose”,
“Calla is a very large, long white flower on thick stalks”. As it becomes apparent
such concepts hardly can be encoded into OWL DL, as they involve so-called
fuzzy or vague concepts, like “creamy”, “dark”, “hot”, “large” and “thick”, for
which a clear and precise definition is not possible (another issue relates to the
representation of terms like “very”, which are called fuzzy concepts modifiers,
as we will see later on).
The problem to deal with imprecise concepts has been addressed several
decades ago by Zadeh [34], which gave birth in the meanwhile to the so-called
fuzzy set and fuzzy logic theory and a huge number of real life applications exists.
Unfortunately, despite the popularity of fuzzy set theory, relative little work
has been carried out in extending DLs towards the representation of imprecise
concepts, notwithstanding DLs can be considered as a quite natural candidate
for such an extension [4, 12, 13, 25, 27, 29, 30, 32, 33] (see also [7], Chapter 6).
In this paper we consider a fuzzy extension of SHOIN (D), the corresponding
DL of the ontology description language OWL DL, and present its syntax and
semantics. The main feature of fuzzy SHOIN (D) is that it allows us to represent
and reason about vague concepts. None of the approaches to fuzzy DLs deal with
the expressive power of the fuzzy extension of SHOIN (D) we present here. Our
purpose is also to integrate most of these contributions within an unique setting
and, thus, hope to define a reference language for fuzzy SHOIN (D). Interesting
features are: (i) concept constructors are based on t-norm, t-conorm, negation
and implication; (ii) concrete domains are fuzzy sets; (iii) fuzzy modifiers are
allowed; and (iv) entailment and subsumption relationships may hold to some
degree in the unit interval [0, 1].
We will proceed as follows. In the following section we recall the description
logic SHOIN (D). In Section 3 we extend SHOIN (D) to the fuzzy case and
discuss some properties of it. Section 4 presents related work, while Section 5
concludes and presents some topics for further research.
2 Preliminaries
The ontology language OWL DL is strictly related to the DL SHOIN (D) [16].
Although several XML and RDF syntaxes for OWL-DL exist, in this paper we
use the traditional description logic notation. For explicating the relationship
between OWL DL and DLs syntax, see e.g. [15, 16]. The purpose of this section
is to make the paper self-contained. More importantly it helps in understanding
1 Taken from a text book on flowers.
2
the differences between classical SHOIN (D) and fuzzy SHOIN (D). The reader
confident with the SHOIN (D) terminology may skip directly to Section 3.
2.1 Syntax
SHOIN (D) allows to reason with concrete data types, such as strings and
integers using so-called concrete domains [2, 21, 22, 23].
Concrete domains. A concrete domain D is a pair 〈∆D, ΦD〉, where ∆D is an
interpretation domain and ΦD is the set of concrete domain predicates d with
a predefined arity n and an interpretation d D ⊆ ∆ n D . For instance, over the
integers ≥20 may be an unary predicate denoting the set of integers greater or
equal to 20. For instance, Person ⊓ ∃age. ≥20 will denote a person whose age
is greater or equal to 20.
Alphabeths. Let C, Ra, Rc, Ia and Ic be non-empty finite and pair-wise disjoint
sets of concepts names, abstract roles names, concrete roles names, abstract
individual names and concrete individual names.
RBox. An abstract role is an abstract role name or the inverse S − of an
abstract role name S (concrete role names do not have inverses). An RBox
R consists of a finite set of transitivity axioms trans(R), and role inclusion
axioms of the form R ⊑ S and T ⊑ U, where R and S are abstract roles, and T
and U are concrete roles. The reflexive-transitive closure of the role inclusion
relationship is denoted with ⊑ ∗ . A role not having transitive sub-roles is called
simple role.
Concepts. The set of SHOIN (D) concepts is defined by the following syntactic
rules, where A is an atomic concept, R is an abstract role, S is an abstract
simple role, Ti are concrete roles, d is a concrete domain predicate, ai and ci
are abstract and concrete individuals, respectively, and n ∈ N:
C −→ ⊤ | ⊥ | A | C1 ⊓ C2 | C1 ⊔ C2 | ¬C |
∀R.C | ∃R.C | (≥ n S) | (≤ n S) | {a1, . . . , an} |
(≥ n T ) | (≤ n T ) | ∀T1, . . . , Tn.D | ∃T1, . . . , Tn.D
D −→ d | {c1, ..., cn}
For instance, we may write the concept
Flower ⊓ (∃hasPetalWidth.(≥20mm ⊓ ≤40mm)) ⊓ ∃hasColour.Red)
to informally denote the set of flowers having petal’s dimension within 20mm
and 40mm, whose colour is red. Here ≥20mm (and ≤40mm) is a concrete domain
predicate. We use (= 1 S) as an abbreviation for (≥ 1 S) ⊓ (≤ 1 S).
3
TBox. A TBox T consists of a finite set of concept inclusion axioms C ⊑ D,
where C and D are concepts. For ease, we use C = D ∈ T in place of C ⊑
D, D ⊑ C ∈ T .
An abstract simple role S is called functional if the interpretation of role S is
always functional (see later for the semantics). A functional role S can always be
obtained from an abstract role by means of the axiom ⊤ ⊑ (≤ 1 S). Therefore,
whenever we say that a role is functional, we assume that ⊤ ⊑ (≤ 1 S) is in the
TBox.
ABox. An ABox A consists of a finite set of concept and role assertion axioms
and individual (in)equality axioms a:C, (a, b):R, (a, c):T , a ≈ b and a �≈ b,
respectively.
Knowledge base. A SHOIN (D) knowledge base K = 〈T , R, A〉 consists of
a TBox T , an RBox R, and an ABox A.
2.2 Semantics
Interpretation. An interpretation I with respect to a concrete domain D is a
pair I = (∆ I , · I ) consisting of a non empty set ∆ I (called the domain), disjoint
from ∆D, and of an interpretation function · I that assigns to each C ∈ C a
subset of ∆ I , to each R ∈ Ra a subset of ∆ I × ∆ I , to each a ∈ Ia an element in
∆ I , to each c ∈ Ic an element in ∆D, to each T ∈ Rc a subset of ∆ I × ∆D and
to each n-ary concrete predicate d the interpretation d D ⊆ ∆ n D .
The mapping · I is extended to concepts and roles as usual:
⊤ I = ∆ I
⊥ I = ∅
(C1 ⊓ C2) I = C1 I ∩ C2 I
(C1 ⊔ C2) I = C1 I ∪ C2 I
(¬C) I = ∆I \ CI (S− ) I = {〈y, x〉: 〈x, y〉 ∈ SI }
(∀R.C) I = {x ∈ ∆I : RI (x) ⊆ CI }
(∃R.C) I = {x ∈ ∆I : RI (x) ∩ CI �= ∅}
(≥ n S) I = {x ∈ ∆I : |SI (x)| ≥ n}
(≤ n S) I = {x ∈ ∆I : |SI (x)| ≤ n}
{a1, . . . , an} I = {a1 I , . . . , an I }
and similarly for the other constructs, where R I (x) = {y: 〈x, y〉 ∈ R I } and |X|
denotes the cardinality of the set X. In particular,
(∃T1, . . . , Tn.d) I = {x ∈ ∆ I : [T1 I (x) × . . . × Tn I (x)] ∩ d D �= ∅} .
4
Satisfiability. The satisfiability of an axiom E in an interpretation I =
(∆ I , · I ), denoted I |= E, is defined as follows: I |= C ⊑ D iff C I ⊆ D I ,
I |= R ⊑ S iff R I ⊆ S I , I |= T ⊑ U iff T I ⊆ U I , I |= trans(R) iff R I is
transitive, I |= a:C iff a I ∈ C I , I |= (a, b):R iff 〈a I , b I 〉 ∈ R I , I |= (a, c):T iff
〈a I , c I 〉 ∈ T I , I |= a ≈ b iff a I = b I , I |= a �≈ b iff a I �= b I .
For a set of axioms E, we say that I satisfies E iff I satisfies each element in
E. If I |= E (resp. I |= E) we say that I is a model of E (resp. E). I satisfies
(is a model of) a knowledge base K = 〈T , R, A〉, denoted I |= K, iff I is a model
of each component T , R and A, respectively.
Logical consequence. An axiom E is a logical consequence of a knowledge
base K, denoted K |= E, iff every model of K satisfies E. According to [15],
the entailment and subsumption problem can be reduced to knowledge base
satisfiability problem (e.g. 〈T , R, A〉 |= a:C iff 〈T , R, A∪{a:¬C}〉 unsatisfiable),
for which decision procedures and reasoning tools exists (e.g. RACER [10] and
FACT [14]).
Example 1 Let us consider the following excerpt of a simple ontology (TBox
T ) about cars, with empty RBox (R = ∅):
Car ⊑ (= 1 maker) ⊓ (= 1 passenger) ⊓ (= 1 speed)
(= 1 maker) ⊑ Car ⊤ ⊑ ∀maker.Maker
(= 1 passenger) ⊑ Car ⊤ ⊑ ∀passenger.N
(= 1 speed) ⊑ Car ⊤ ⊑ ∀speed.Km/h
Roadster ⊑ Cabriolet ⊓ ∃passenger.{2}
Cabriolet ⊑ Car ⊓ ∃topType.SoftTop
SportsCar = Car ⊓ ∃speed.≥ 245km/h
In T , the value for speed ranges over the concrete domain of kilometres per
hour, Km/h, while the value for passengers ranges over the concrete domain
of natural numbers, N. The concrete predicate ≥ 245km/h is true if the value is
greater or equal than to 245km/h.
The ABox A contains the following assertions:
mgb:Roadster ⊓ (∃maker.{mg}) ⊓ (∃speed.≤ 170km/h)
enzo:Car ⊓ (∃maker.{ferrari}) ⊓ (∃speed.> 350km/h)
tt:Car ⊓ (∃maker.{audi}) ⊓ (∃speed.= 243km/h)
Consider the knowledge base K = 〈T , R, A〉. It is then easily verified that, e.g.
K |= Roadster ⊑ Car K |= mg:Maker
K |= enzo:SportsCar K |= tt:¬SportsCar .
5
✷
The above example illustrates an evident difficulty in defining the class of sport
cars. Indeed, it is highly questionable why a car whose speed is 243km/h is not
a sport car any more. The point is that essentially, the higher the speed the
more likely a car is a sports car, which makes the concept of sports car rather
a fuzzy concept, i.e. vague concept, rather than a crisp one. In the next section
we will see how to represent such concepts more appropriately.
3 Fuzzy OWL DL
Fuzzy sets have been introduced by Zadeh [34] as a way to deal with vague
concepts like low pressure, high speed and the like. Formally, a fuzzy set A
with respect to a universe X is characterized by a membership function µA :
X → [0, 1], assigning an A-membership degree, µA(x), to each element x in X.
µA(x) gives us an estimation of the belonging of x to A. Typically, if µA(x) = 1
then x definitely belongs to A, while µA(x) = 0.8 means that x is “likely” to be
an element of A.
When we switch to fuzzy logics, the notion of degree of membership µA(x)
of an element x ∈ X w.r.t. the fuzzy set A over X is regarded as the degree of
truth in [0, 1] of the statement “x is A”. Accordingly, in our fuzzy DL, (i) a
concept C, rather than being interpreted as a classical set, will be interpreted
as a fuzzy set and, thus, concepts become imprecise; and, consequently, (ii) the
statement “a is C”, i.e. a:C, will have a truth-value in [0, 1] given by the degree
of membership of being the individual a a member of the fuzzy set C.
In the following, we present first some preliminaries on fuzzy set theory (for
a more complete and comprehensive presentation see e.g. [6, 11, 17, 8]) and then
define fuzzy SHOIN (D).
3.1 Preliminaries on fuzzy set theory
Let X be a crisp set and let A be a fuzzy subset of X, with membership function
µA(x), or simply A(x) ∈ [0, 1], x ∈ X. The support of A, supp(A), is the crisp
set supp(A) = {x ∈ X: A(x) �= 0}. The scalar cardinality of A, |A|, is defined
as |A| = �
x∈X
A(x). The fuzzy powerset of X, F(X), is the set of all the
fuzzy sets over X. Let A, B ∈ F(X). We say that A and B are equivalent iff
A(x) = B(x), ∀x ∈ X. A is a crisp subset of B iff A(x) ≤ B(x), ∀x ∈ X. Note
that either A is a subset of B or it is not. We will see later on a different notion
of subset, in which A is a subset of B to some degree in [0, 1]. We next give the
basic definitions on fuzzy set operations (complement, intersection and union).
The complement of A, ¬A, is given by membership function (¬A)(x) =
n(A(x)), for any x ∈ X. The function n: [0, 1] → [0, 1], called negation, has to
satisfy the following conditions and extends boolean negation:
• n(0) = 1 and n(1) = 0;
• ∀a, b ∈ [0, 1], a ≤ b implies n(b) ≤ n(a);
• ∀a ∈ [0, 1], n(n(a)) = a.
6
Several negation functions have been given in the literature, e.g. Lukasiewicz
negation nL(a) = 1−a (syntax, ¬L) and Gödel negation nG(0) = 1 and n(a) = 0
if a > 0 (syntax, ¬G).
The intersection of two fuzzy sets A and B is given (A∧B)(x) = t(A(x), B(x)),
where t is a triangular norm, or simply t-norm. A t-norm is, called conjunction,
is a function t: [0, 1] × [0, 1] → [0, 1] that has to satisfy the following conditions:
• ∀a ∈ [0, 1], t(a, 1) = a;
• ∀a, b, c ∈ [0, 1], b ≤ c implies t(a, b) ≤ t(a, c);
• ∀a, b ∈ [0, 1], t(a, b) = t(b, a);
• ∀a, b, c ∈ [0, 1], t(a, t(b, c)) = t(t(a, b), c).
Examples of t-norms are: tL(a, b) = max(a + b − 1, 0) (Lukasiewicz t-norm,
syntax ∧L), tG(a, b) = min(a, b) (Gödel t-norm, syntax ∧G), and tP (a, b) = a · b
(product t-norm, syntax ∧P ). Note that ∀a ∈ [0, 1], t(a, 0) = 0.
The union of two fuzzy sets A and B is given (A ∨ B)(x) = s(A(x), B(x)),
where s is a triangular co-norm, or simply s-norm. A s-norm, called disjunction,
is a function s: [0, 1] × [0, 1] → [0, 1] that has to satisfy the following conditions:
• ∀a ∈ [0, 1], s(a, 0) = a;
• ∀a, b, c ∈ [0, 1], b ≤ c implies s(a, b) ≤ s(a, c);
• ∀a, b ∈ [0, 1], s(a, b) = s(b, a);
• ∀a, b, c ∈ [0, 1], s(a, s(b, c)) = s(s(a, b), c).
Examples of s-norms are: sL(a, b) = min(a + b, 1) (Lukasiewicz s-norm, syntax
∨L), sG(a, b) = max(a, b) (Gödel s-norm, syntax ∨G), and sP (a, b) = a+b−a·b
(product s-norm, syntax ∨P ). Note that if we consider Lukasiewicz negation,
then Lukasiewicz, Gödel and product s-norm are related to their respective tnorm
according to the De Morgan law: ∀a, b ∈ [0, 1], s(a, b) = n(t(n(a), n(b))).
Another important operator is implication, denoted →, that gives a truthvalue
to the formula A → B, when the truth of A and B are known. A fuzzy
implication is a function i: [0, 1] × [0, 1] → [0, 1] that has to satisfy the following
conditions:
• ∀a, b, c ∈ [0, 1], a ≤ b implies i(a, c) ≥ i(b, c);
• ∀a, b, c ∈ [0, 1], b ≤ c implies i(a, b) ≤ i(a, c);
• ∀a ∈ [0, 1], i(0, b) = 1;
• ∀a ∈ [0, 1], i(a, 1) = 1;
• i(1, 0) = 0.
7
In classical logic, a → b is a shorthand for ¬a∨b. A generalization to fuzzy logic
is, thus, ∀a, b ∈ [0, 1], i(a, b) = s(n(a), b). For instance, ∀a, b ∈ [0, 1], iKD(a, b, ) =
max(1 − a, b) is the so-called Kleene-Dienes implication (syntax, →KD). Another
approach to fuzzy implication is based on the so-called residuum. His
formulation starts from the fact that in classical logic ¬a ∨ b can be re-written
as max{c ∈ {0, 1}: a ∧ c ≤ b}. Therefore, another generalization of implication
to fuzzy logic is
∀a, b ∈ [0, 1], i(a, b) = sup{c ∈ [0, 1]: t(a, c) ≤ b} .
For residuum based implication, i(a, b) = 1 if a ≤ b. If a > b then, according
to the chosen t-norm, we have that e.g. iL(a, b) = 1 − a + b for Lukasiewicz
implication (syntax, →L), iG(a, b) = b for Gödel implication (syntax, →G)) and
iP (a, b) = a/b for product implication (syntax, →P ). Note that, for Lukasiewcz
implication, s-norm and negation, we have iL(a, b) = sL(nL(a), b). The same
holds using Kleene-Dienes implication, Lukasiewicz negation and Gödel s-norm.
On the other hand iP (a, b) �= sP (nG(a), b) (for instance, for 0 < a ≤ b < 1,
iP (a, b) = 1, while sP (nG(a), b) = b < 1).
Another interesting question is when ∀a, b ∈ [0, 1], i(a, b) = n(t(a, n(b))
holds, which in formulae is formulated as a → b ≡ ¬(a ∧ ¬b). It turns out
that e.g., in Zadeh’s logic [34] (i.e. using →KD, ∧G, ¬L) this relation holds. It
holds as well in the so-called Lukasiewcz logic (i.e. using →L, ∧L, ¬L), while it
does neither hold for Gödel logic (i.e. using →G, ∧G, ¬G) nor for the product
logic (i.e. using →P , ∧P , ¬G). For them, just consider the case 1 > a > b > 0 to
verify the inequality. We will see later on that whenever i(a, b) �= n(t(a, n(b))
then under the fuzzy semantics, ∀R.C is not equivalent to ¬∃R.¬C.
Fuzzy implication can also be used to determine the degree of subset relationship
between two fuzzy subsets A and B over X. Indeed, we define the degree
of subsumption between A and B, denoted A → B, as infx∈X i(A(x), B(x)),
where i is an implication function. Note that if ∀x ∈ [0, 1], A(x) ≤ B(x) holds
then A → B evaluates to 1. Of course, it may be that A → B evaluates to a
value 0 < v < 1 as well.
We conclude the discussion on fuzzy implication by noting that we have the
following inferences: assume a ≥ n and i(a, b) ≥ m. Then
• under Kleene-Dienes implication we infer that if n > 1 − m then b ≥ m.
Indeed, from i(a, b) = max(1 − a, b) ≥ m, either 1 − a ≥ m or b ≥ m. But
a ≥ n, so 1 − a ≥ m implies 1 − m ≥ a ≥ n > 1 − m, a contradiction.
Therefore, b ≥ m must hold. This has been used in [29].
• under residuum based implication w.r.t. a t-norm t, we infer that b ≥
t(n, m). Indeed, from i(a, b) = sup{c: t(a, c) ≤ b} ≥ m and a ≥ n we have
t(n, m) ≤ t(n, c) ≤ t(a, c) ≤ b.
A (binary) fuzzy relation R over two crisp sets X and Y is a function R: X×Y →
[0, 1]. The inverse of R is the function R −1 : Y × X → [0, 1] with membership
function R −1 (y, x) = R(x, y), for every x ∈ X and y ∈ Y . The composition
8
of two fuzzy relations R1: X × Y → [0, 1] and R2: Y × Z → [0, 1] is defined
as (R1 ◦ R2)(x, z) = sup y∈Y t(R1(x, y), R2(y, z)), where t is a t-norm. A fuzzy
relation R is said to be transitive iff (R ◦ R)(x, z) ≤ R(x, z).
We conclude this part with fuzzy modifiers. Fuzzy modifiers applies to fuzzy
sets to change their membership function. Well known examples are modifiers
like very, more or less, slightly, etc. These allow us to define fuzzy sets
like very(High) and slightly(Mature). Formally, a modifier, fm, is a function
fm: [0, 1] → [0, 1]. For instance, we may define very(x) = x 2 , while define
slightly(x) = √ x.
In the following, we use ∧, ∨, ¬ and → in infix notation, in place of a t-norm
t, s-norm s, negation n and implication operator i.
3.2 Fuzzy SHOIN (D)
In this section we give syntax and semantics of fuzzy SHOIN (D), using the
fuzzy operators defined in the previous section. We generalize the semantics
given in [13, 29, 32].
3.2.1 Syntax
We have seen that SHOIN (D) allows to reason with concrete data types, such
as strings and integers using so-called concrete domains. In our fuzzy approach,
concrete domains may be based on fuzzy sets as well.
Concrete fuzzy domain. A concrete fuzzy domain is a pair 〈∆D, ΦD〉, where
∆D is an interpretation domain and ΦD is the set of concrete fuzzy domain
predicates d with a predefined arity n and an interpretation d D : ∆ n D → [0, 1],
which is a n-ary fuzzy relation over ∆D.
For instance, as for SHOIN (D), the predicate ≤18 may be an unary crisp
predicate over the natural numbers denoting the set of integers smaller or equal
to 18, i.e. ≤18: Natural → [0, 1] and
So,
≤18(x) =
� 1 if x ≤ 18
0 otherwise .
Minor = Person ⊓ ∃age. ≤18
defines a person, whose age is less or equal 18, i.e. it defines a minor.
On the other hand, concerning non crisp fuzzy domain predicates, we recall
that in fuzzy set theory and practice there are many membership functions for
fuzzy sets membership specification. However, the triangular, the trapezoidal,
the L-function and the R-function are simple, yet most frequently used to specify
membership degrees. The functions are defined over the set of non-negative reals
R + ∪ {0}. The trapezoidal function, trz(x, a, b, c, d), is defined as follows: let
9
(1)
(a) (b)
(c) (d)
Figure 1: (a) Trapezoidal function; (b) Triangular function; (c) L-function; (d)
R-function
a < b ≤ c < d be rational numbers then (see Figure 1)
⎧
0
⎪⎨ (x − a)/(b − a)
if x ≤ a
if x ∈ [a, b]
trz(x; a, b, c, d) = 1
⎪⎩
(d − x)/(d − c)
0
if x ∈ [b, c]
if x ∈ [c, d]
if x ≥ d .
A triangular function, tri(x; a, b, c), is such that (see Figure 1)
⎧
⎪⎨
0 if x ≤ a
(x − a)/(b − a) if x ∈ [a, b]
tri(x; a, b, c) =
⎪⎩
(c − x)/(c − b) if x ∈ [b, c]
0 if x ≥ c .
Note that tri(x; a, b, c) = trz(x; a, b, b, c).
The L-function is defined as (see Figure 1)
⎧
⎨ 1 if x ≤ a
L(x; a, b) = (b − x)/(b − a) if x ∈ [a, b]
⎩
0 if x ≥ b .
10
Finally, the R-function is defined as (see Figure 1)
⎧
⎨ 0 if x ≤ a
R(x; a, b) = (x − a)/(b − a)
⎩
1
if x ∈ [a, b]
if x ≥ b .
Using these functions, we may then define, for instance, Young: Natural → [0, 1]
to be a fuzzy concrete predicate over the natural numbers denoting the degree
of youngness of a person’s age. The concrete fuzzy predicate Young may be
defined as
Young(x) = L(x; 10, 30) .
So,
will denote a young person.
YoungPerson = Person ⊓ ∃age.Young (2)
Modifiers. We allow modifiers in fuzzy SHOIN (D). Fuzzy modifiers, like
very, more or less and slightly, apply to fuzzy sets to change their membership
function. Formally, a modifier is a function fm: [0, 1] → [0, 1]. For instance,
we may define very(x) = x 2 , while define slightly(x) = √ x. Modifiers has
been considered, for instance, in [13, 32]. From a syntax point of view, if M is a
new alphabet for modifier symbols, m ∈ M is a modifier and C is a SHOIN (D)
concept, then m(C) is fuzzy SHOIN (D) concept as well. For instance, by
referring to Example 1, we may define the concept of sports car as the concept
SportsCar = Car ⊓ ∃speed.very(High) , (3)
where very is a concept modifier, with membership function very(x) = x 2 ,
and High is a fuzzy concrete predicate over the domain of speed expressed in
kilometres per hour and may be defined as
High(x) = R(x; 80, 250) .
Similarly, we may represent “Calla is a very large, long white flower on thick
stalks” as
Calla = Flower ⊓ (∃hasSize.very(Large)) ⊓ (∃hasPetalWidth.Long)⊓
⊓(∃hasColour.White) ⊓ (∃hasStalks.Thick) ,
where Large, Long and Thick are fuzzy concrete predicates.
In summary, the syntax of fuzzy SHOIN (D) concepts is as follows:
C −→ ⊤ | ⊥ | A | C1 ⊓ C2 | C1 ⊔ C2 | ¬C | m(C)
∀R.C | ∃R.C | (≥ n S) | (≤ n S) | {a1, . . . , an} |
(≥ n T ) | (≤ n T ) | ∀T1, . . . , Tn.D | ∃T1, . . . , Tn.D
D −→ d | {c1, ..., cn}
Concerning axioms and assertions, similarly to [29], we introduce fuzzy axioms.
Let be n ∈ (0, 1].
11
Fuzzy RBox. A fuzzy RBox R is a finite set of SHOIN (D) transitivity
axioms trans(R) and fuzzy role inclusion axioms of the form 〈α ≥ n〉, 〈α ≤ n〉,
〈α > n〉 and 〈α > n〉, where α is a SHOIN (D) role inclusion axiom.
Fuzzy TBox. A fuzzy TBox T consists of a finite set of fuzzy concept inclusion
axioms of the form 〈α ≥ n〉, 〈α ≤ n〉, 〈α > n〉 and 〈α < n〉 where α is a
SHOIN (D) concept inclusion axiom;
Fuzzy ABox. A fuzzy ABox A consists of a finite set of fuzzy concept and
fuzzy role assertion axioms of the form 〈α ≥ n〉, 〈α ≤ n〉, 〈α > n〉, or 〈α < n〉,
where α is a SHOIN (D) concept or role assertion. As for the crisp case, A
may also contain a finite set of individual (in)equality axioms a ≈ b and a �≈ b,
respectively.
For instance, 〈a:C ≥ 0.1〉, 〈(a, b):R ≤ 0.3〉, 〈R ⊑ S ≥ 0.4〉, or 〈C ⊑ D ≤ 0.6〉
are fuzzy axioms. Informally, from a semantics point of view, a fuzzy axiom
〈α ≤ n〉 constrains the membership degree of α to be less or equal to n (similarly
for ≥, >, <). As a consequence,
〈jim:YoungPerson ≥ 0.2〉 ,
i.e. 〈jim:Person ⊓ ∃age.Young ≥ 0.2〉, dictates that jim is a YoungPerson with
degree at least 0.2.
On the other hand, a fuzzy concept inclusion axiom of the form
〈C ⊑ D ≥ n〉
dictates that the subsumption degree between C and D is at least n.
Fuzzy knowledge base. A SHOIN (D) fuzzy knowledge base K = 〈T , R, A〉
consists of a fuzzy TBox T , a fuzzy RBox R, and a fuzzy ABox A.
3.2.2 Semantics
The semantics extends [29]. The main idea is that concepts and roles are interpreted
as fuzzy subsets of an interpretation’s domain. Therefore, SHOIN (D)
axioms, rather being satisfied (true) or unsatisfied (false) in an interpretation,
become a degree of truth in [0, 1].
Fuzzy interpretation. A fuzzy interpretation I with respect to a concrete
domain D is a pair I = (∆ I , · I ) consisting of a non empty set ∆ I (called the
domain), disjoint from ∆D, and of a fuzzy interpretation function · I that assigns
• to each abstract concept C ∈ C a function C I : ∆ I → [0, 1];
• to each abstract role R ∈ Ra a function R I : ∆ I × ∆ I → [0, 1];
12
• to each abstract functional role R ∈ Ra a partial function R I : ∆ I × ∆ I →
[0, 1] such that for all x ∈ ∆ I there is an unique y ∈ ∆ I on which R I (x, y)
is defined;
• to each abstract individual a ∈ Ia an element in ∆ I ;
• to each concrete individual c ∈ Ic an element in ∆D;
• to each concrete role T ∈ Rc a function R I : ∆ I × ∆D → [0, 1];
• to each concrete functional role T ∈ Rc a partial function t I : ∆ I × ∆D →
[0, 1] such that for all x ∈ ∆ I there is an unique v ∈ ∆D on which T I (x, v)
is defined;
• to each modifier m ∈ M the modifier function fm: [0, 1] → [0, 1];
• to each n-ary concrete fuzzy predicate d the fuzzy relation d D : ∆ n D → [0, 1].
The mapping · I is extended to concepts and roles as specified in the following
table (where x, y ∈ ∆ I , v ∈ ∆D):
⊤ I (x) = 1
⊥ I (x) = 0
(C1 ⊓ C2) I (x) = C1 I (x) ∧ C2 I (C1 ⊔ C2)
(x)
I (x) = C1 I (x) ∨ C2 I (¬C)
(x)
I (x) = ¬CI (x))
(m(C)) I (x) = fm(C I (∀R.C)
(x))
I (x) = infy∈∆I RI (x, y) → CI (∃R.C)
(y)
I (x) = supy∈∆I RI (x, y) ∧ CI (≥ n S)
(y)
I (x) =
�n sup I
{y1, . . . , yn} ⊆ ∆ i=1 SI (x, yi)
|{y1, . . . , yn}| = n
(≤ n S) I (x) = ¬(≥ n + 1 S) I (x)
{a1, . . . , an} I (x) = � n
i=1 ai I = x
d(v) = d D (v)
{c1, . . . , cn} I (v) = � n
i=1 ci I = v
(∀T1, . . . , Tn.D) I (x) = inf y1,...,yn∈∆D I ( � n
i=1 Ti I (x, yi)) → D I (y1, . . . , yn)
(∃T1, . . . , Tn.D) I (x) = sup y1,...,yn∈∆D I ( � n
i=1 Ti I (x, yi)) ∧ D I (y1, . . . , yn)
(S − ) I (x, y) = S I (y, x) .
We comment briefly some points. The semantics of ∃R.C
(∃R.C) I (d) = sup y∈∆ I R I (x, y) ∧ C I (y)
is the result of viewing ∃R.C as the open first order formula ∃y.FR(x, y)∧FC(y)
(where F is the obvious translation of roles and concepts into First-Order Logic -
FOL) and the existential quantifier ∃ is viewed as a disjunction over the elements
of the domain. Similarly,
(∀R.C) I (x) = inf y∈∆ I R I (x, y) → C I (y)
13
is related to the open first order formula ∀y.FR(x, y) → FC(y), where the universal
quantifier ∀ is viewed as a conjunction over the elements of the domain.
However, as we already pointed out in Section 3.1, unlike the classical case, in
general we do not have that (∀R.C) I = (¬∃R.¬C) I . For instance, it holds in
Lukasiewicz logic, but not in Gödel logic. Also interesting is that (see [12]) the
axiom ⊤ ⊑ ¬(∀R.A) ⊓ (¬∃R.¬A) has no classical model. However, in [12] it is
shown that in Gödel logic it has no finite model, but has an infinite model.
Another point concerns the semantics of number restrictions. The semantics
of the concept (≥ n S)
(≥ n S) I (x) = sup {y1, . . . , yn} ⊆ ∆ I
|{y1, . . . , yn}| = n
� n
i=1 SI (x, yi)
is the result of viewing (≥ n S) as the open first order formula
∃y1, . . . , yn.
n�
S(x, yi) ∧ �
i=1
1≤i<j≤n
yi �= yj .
That is, there are at least n distinct elements that satisfy to some degree S(x, yi).
This guarantees us that ∃S.⊤ ≡ (≥ 1 S). The semantics of (≤ n S) is defined
in such a way to guarantee the classical relationship (≤ n S) ≡ ¬(≥ n + 1 S).
An alternative definition for the (≥ n S) and the (≤ n S) constructs may
rely on the scalar cardinality of a fuzzy set. However, we prefer to stick on the
formulation, which derives directly from its FOL translation.
Finally, the mapping · I is extended to non-fuzzy axioms as specified in the
following table (where a, b ∈ Ia):
(R ⊑ S) I = inf x,y∈∆ I R I (x, y) → S I (x, y)
(T ⊑ U) I = inf x,y∈∆ I T I (x, y) → U I (x, y)
(C ⊑ D) I = inf x∈∆ I C I (x) → D I (x)
(a:C) I = C I (a I )
((a, b):R) I = R I (a I , b I ) .
Note here that e.g. the semantics of a concept inclusion axiom C ⊑ D is derived
directly from its FOL translation, which is of the form ∀x.FC(x) → FD(x).
This definition is novel and is clearly different from the approaches in which
C ⊑ D is viewed as ∀x.C(x) ≤ D(x). This latter approach has the effect that
the subsumption relationship is a classical {0, 1} relationship, while the former
has the advantage that subsumption is determined up to a certain degree in
[0, 1].
Satisfiability. The notion of satisfiability of a fuzzy axiom E by a fuzzy
interpretation I, denoted I |= E, is defined as follows: I |= trans(R), iff
∀x, y ∈ ∆ I .R I (x, y) = sup z∈∆ I R I (x, z) ∧ R I (z, y). I |= 〈α ≥ n〉, where α is a
role inclusion or concept inclusion axiom, iff α I ≥ n. Similarly, for the other
relations ≤, < and >. I |= 〈α ≥ n〉, where α is a concept or a role assertion
14
axiom, iff α I ≥ n. Similarly, for the other relations ≤, <, >. Finally, I |= a ≈ b
iff a I = b I and I |= a �≈ b iff a I �= b I .
For a set of fuzzy axioms E, we say that I satisfies E iff I satisfies each
element in E. If I |= E (resp. I |= E) we say that I is a model of E (resp. E). I
satisfies (is a model of) a fuzzy knowledge base K = 〈T , R, A〉, denoted I |= K,
iff I is a model of each component T , R and A, respectively.
Logical consequence. A fuzzy axiom E is a logical consequence of a knowledge
base K, denoted K |= E iff every model of K satisfies E.
The interesting point is that according to our semantics, e.g. a minor is a
young person to a certain degree and is obtained without explicitly mentioning
it. This inference can not be achieved in classical SHOIN (D). Similarly, by
referring to Example 1, we will have that the car tt will be a sports car to a
certain degree. Therefore, unlike Example 1, tt is now likely a sport car, as it
should be. The following two examples highlight these points.
Example 2 Let us consider Example 1, where all axioms of the TBox and ABox
are asserted with degree 1, i.e. are of the form 〈α ≥ 1〉. We replace the definition
of SportsCar with Definition (3). Then we have that (interpreting conjunction
as min)
K |= 〈SportsCar ⊑ Car ≥ 1〉 K |= 〈mgb:SportsCar ≤ 0.25〉
K |= 〈enzo:SportsCar ≥ 1〉 K |= 〈tt:SportsCar ≥ 0.82〉 .
Note how the maximal speed limit of the mgb car (≤ 170km/h) induces an upper
limit, 0.25, of the membership degree. Neither this inference is possible in classical
SHOIN (D), nor the one involving tt.
Example 3 Consider the knowledge base K with Definitions (1) and (2). Then
under Lukasiewicz logic we have that (see [31])
K |= 〈Minor ⊑ YoungPerson ≥ 0.2〉 ,
which is a relationship not captured with classical SHOIN (D).
Best truth value bound. Finally, given K and an axiom α, where α is neither
a transitivity axiom, nor an individual (in) equality axiom, it is of interest to
compute α’s best lower and upper degree value bounds. The greatest lower
bound of α w.r.t. K (denoted glb(K, α)) is
glb(K, α) = sup{n: K |= 〈α ≥ n〉} ,
while the least upper bound of α with respect to K (denoted lub(K, α) is
lub(K, α) = inf{n: K |= 〈α ≤ n〉} ,
15
where sup ∅ = 0 and inf ∅ = 1. Determining the lub and the glb is called the Best
Degree Bound (BDB) problem. For instance, the entailments in Examples 2 and
3 are the best possible degree bounds. Furthermore, note that,
lub(Σ, a:C) = ¬glb(Σ, a:¬C) , (4)
i.e. the lub can be determined through the glb (and vice-versa). Similarly,
lub(Σ, (a, b):R) = ¬glb(Σ, a:¬∃R.{b}) holds. Also, note that, Σ |= 〈α ≥ n〉
iff glb(Σ, α) ≥ n, and similarly Σ |= 〈α ≤ n〉 iff lub(Σ, α) ≤ n hold.
Concerning the entailment problem, it is quite easily verified that, as for
the crisp case, the entailment problem can be reduced to the unsatisfiability
problem:
〈T , R, A〉 |= 〈α ≥ n〉 iff 〈T , R, A ∪ {〈α < n〉}〉 is not satisfiable
〈T , R, A〉 |= 〈α ≤ n〉 iff 〈T , R, A ∪ {〈α > n〉}〉 is not satisfiable .
3.3 Reasoning
Unfortunately, from a computational point of view, no calculus exists yet checking
satisfiability of fuzzy SHOIN (D) knowledge bases. [13, 32] report a calculus
for the case of ALC [26] (with concept constructors ⊤, ⊥, ¬, ⊓, ⊔, ∀, ∃) with modifiers
and simple TBox, with min, max and →KD connectives. No indication for
the BDB problem is given. [27, 29] reports a calculus for ALC and simple TBox,
with min, max and →KD connectives and addresses the BDB problem and, [30]
shows how the satisfiability problem and the BDB problem can be reduced to
classical ALC and, thus, can be resolved by means of a tools like FACT and
RACER. However, despite these negative results, recently [31] reports a calculus
for ALC(D) whenever the connectives, the modifiers and the concrete fuzzy
predicates are representable as a bounded Mixed Integer Program. For instance,
Lukasiewicz logic satisfies these conditions as well as the membership functions
for concrete fuzzy predicates we have presented in this paper. Additionally,
modifiers should be a combination of linear functions. In that case the calculus
consists of a set of constraint propagation rules and an invocation to an oracle
for bounded Mixed Integer Programming. But, indeed, the computational
aspect is definitely a point that has to be addressed in forthcoming works.
4 Related work
Several ways of extending DL using the theory of fuzzy logic have been proposed
in the literature. The first work is due to Yen [33] who considered a
sub-language of ALC, FL − [5]. However, it already informally talks about the
use of modifiers and concrete domains. Though, the unique reasoning facility,
the subsumption test, is a crisp yes/no question. Tresp [32] considered fuzzy
ALC extended with a special form of modifiers, which are a combination of two
linear functions. min, max and 1−x membership functions has been considered
16
and a sound and complete reasoning algorithm testing the subsumption relationship
has been presented. Similar to our approach, a linear programming oracle
is needed. Assertional reasoning has been considered by Straccia [27, 28, 29],
where fuzzy assertion axioms have been allowed in fuzzy ALC (with min, max
and 1 − x functions), concept modifiers are not allowed however ([28] reports a
four-valued variant of fuzzy ALC). He also introduced the BDB problem and
provided a sound and complete reasoning algorithm based on completion rules
([30] provides a translation of fuzzy ALC into classical ALC). For an application
see [24]. In the same spirit [13] extend Straccia’s fuzzy ALC with concept
modifiers of the form fm(x) = x β , where β > 0. A sound and complete reasoning
algorithm for the graded subsumption problem, based on completion rules,
is presented. Finally, [25] start addressing the issue of alternative semantics of
quantifiers in fuzzy ALC (without the assertional component). No reasoning
algorithm is given. Concerning [31], we already addressed it in the previous
section. Finally, [12] considers ALC under arbitrary t-norm and reports, among
others, a procedure deciding |= 〈C ⊑ D ≥ 1〉 and deciding whether 〈C ⊑ D ≥ 1〉
is satisfiable, by a reduction to the propositional BL logic.
5 Conclusions and outlook
We have presented a fuzzy extension of SHOIN (D) showing that its representation
and reasoning capabilities go clearly beyond classical SHOIN (D).
Interestingly, we allow modifiers, fuzzy concrete domain predicates and fuzzy
axioms to appear in a SHOIN (D) knowledge base and the entailment and the
subsumption relationship hold to a certain degree. To the best of our knowledge,
no other work has yet extended the semantics to SHOIN (D) in such a
way. The argument supporting the necessity of such an extension relies on the
fact that vague concepts are abundant in human knowledge and, thus, appear
likely in Web content.
The main direction for future work involves the computational aspect. Currently,
we are addressing the fundamental issue to develop a calculus for reasoning
within SHOIN (D), extending [31].
Another direction is in extending fuzzy SHOIN (D) with fuzzy quantifiers
(see [19, 20] for an overview on fuzzy quantifiers), where the ∀ and ∃ quantifiers
are replaced with fuzzy quantifiers like most, some, usually and the like (see
[25] for a preliminary work in this direction). This allows to define concepts like
TopCustomer = Customer ⊓ (Usually)buys.ExpensiveItem
ExpensiveItem = Item ⊓ ∃price.High .
Here, the fuzzy quantifier Usually replaces the classical quantifier ∀ and High is
a fuzzy concrete predicate. Fuzzy quantifiers can be applied to inclusion axioms
as well, allowing to express, e.g.
(Most)Bird ⊑ FlyingObject .
17
Here the fuzzy quantifier Most replaces the classical universal quantifier ∀ assumed
in the inclusion axioms. The above axiom allows to state that most birds
fly.
Ultimately, we believe that the fuzzy extension of SHOIN (D) is of great
interest to the Semantic Web community, as it allows to express naturally a
wide range of concepts of actual domains, for which a classical SHOIN (D)
representation is unsatisfactory.
References
[1] Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, and
Peter F. Patel-Schneider, editors. The Description Logic Handbook: Theory,
Implementation, and Applications. Cambridge University Press, 2003.
[2] Franz Baader and Philipp Hanschke. A schema for integrating concrete domains
into concept languages. In Proceedings of the 12th International Joint
Conference on Artificial Intelligence (IJCAI-91), pages 452–457, Sydney,
1991.
[3] T. Berners-Lee, J. Hendler, and O. Lassila. The semantic web. The Scientific
American, 284(5):34–43, 2001.
[4] P. Bonatti and A. Tettamanzi. Some complexity results on fuzzy description
logics. In A. Petrosino V. Di Gesù, F. Masulli, editor, WILF 2003 International
Workshop on Fuzzy Logic and Applications, LNCS 2955, Berlin,
2004. Springer Verlag.
[5] Ronald J. Brachman and Hector J. Levesque. The tractability of subsumption
in frame-based description languages. In Proceedings of AAAI-84, 4th
Conference of the American Association for Artificial Intelligence, pages
34–37, Austin, TX, 1984. [a] An extended version appears as [18].
[6] Didier Dubois and Henri Prade. Fuzzy Sets and Systems. Academic Press,
New York, NJ, 1980.
[7] Pan et al. Specification of coordination of rule and ontology languages.
Technical report, Knowledgeweb Network of Excellence, EU-IST-2004-
507482, 2004. Deliverable D2.5.1.
[8] S. Gottwald. A Treatise on Many-Valued Logics. A Research Studies Press
Book, 2000.
[9] N. Guarino and R. Poli. Formal ontology in conceptual analysis and knowledge
representation. International Journal of Human and Computer Studies,
43(5/6):625–640, 1995.
[10] Volker Haarslev and Ralf Möller. RACER system description. In Proceedings
of International Joint Conference on Automated Reasoning (IJCAR-
01), number 2083 in LNAI, pages 701–705, Siene, Italy, 2001. Springer.
18
[11] Petr Hájek. Metamathematics of Fuzzy Logic. Kluwer, 1998.
[12] Petr Hájek. Making fuzzy description logics more expressive. Fuzzy Sets
and Systems, 2005. To appear.
[13] Steffen Hölldobler, Hans-Peter Störr, and Tran Dinh Khang. The subsumption
problem of the fuzzy description logic ALCF H. In Proceedings of the
10th International Conference on Information Processing and Managment
of Uncertainty in Knowledge-Based Systems, (IPMU-04), 2004.
[14] Ian Horrocks. Using an expressive description logic: Fact or fiction? In
Proc. of the 8th Int. Conf. on the Principles of Knowledge Representation
and Reasoning (KR-98), 1998.
[15] Ian Horrocks and Peter Patel-Schneider. Reducing OWL entailment to
description logic satisfiability. Journal of Web Semantics, 2004. To Appear.
[16] Ian Horrocks, Peter F. Patel-Schneider, and Frank van Harmelen. From
SHIQ and RDF to OWL: The making of a web ontology language. Journal
of Web Semantics, 1(1):7–26, 2003.
[17] George J. Klir and Bo Yuan. Fuzzy sets and fuzzy logic: theory and applications.
Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1995.
[18] Hector J. Levesque and Ronald J. Brachman. Expressiveness and tractability
in knowledge representation and reasoning. Computational Intelligence,
3:78–93, 1987.
[19] Yaxin Liu and Etienne E. Kerre. An overview of fuzzy quantifiers. (i).
interpretations. Fuzzy Sets Syst., 95(1):1–21, 1998.
[20] Yaxin Liu and Etienne E. Kerre. An overview of fuzzy quantifiers (ii).
reasoning and applications. Fuzzy Sets Syst., 95(2):135–146, 1998.
[21] C. Lutz. Description logics with concrete domains—a survey. In Advances
in Modal Logics Volume 4. King’s College Publications, 2003.
[22] Carsten Lutz. Reasoning with concrete domains. In Proceedings of the
Sixteenth International Joint Conference on Artificial Intelligence, pages
90–95. Morgan Kaufmann Publishers Inc., 1999.
[23] Carsten Lutz. Nexp time-complete description logics with concrete domains.
ACM Trans. Comput. Logic, 5(4):669–705, 2004.
[24] Carlo Meghini, Fabrizio Sebastiani, and Umberto Straccia. A model of multimedia
information retrieval. Journal of the ACM, 48(5):909–970, 2001.
[25] D. Sánchez and G.B. Tettamanzi. Generalizing quantification in fuzzy description
logics. In Proceedings 8th Fuzzy Days in Dortmund, 2004.
19
[26] Manfred Schmidt-Schauß and Gert Smolka. Attributive concept descriptions
with complements. Artificial Intelligence, 48:1–26, 1991.
[27] Umberto Straccia. A fuzzy description logic. In Proc. of the 15th Nat.
Conf. on Artificial Intelligence (AAAI-98), pages 594–599, Madison, USA,
1998.
[28] Umberto Straccia. A framework for the retrieval of multimedia objects
based on four-valued fuzzy description logics. In F. Crestani and Gabriella
Pasi, editors, Soft Computing in Information Retrieval: Techniques and
Applications, pages 332–357. Physica Verlag (Springer Verlag), Heidelberg,
Germany, 2000.
[29] Umberto Straccia. Reasoning within fuzzy description logics. Journal of
Artificial Intelligence Research, 14:137–166, 2001.
[30] Umberto Straccia. Transforming fuzzy description logics into classical description
logics. In Proceedings of the 9th European Conference on Logics in
Artificial Intelligence (JELIA-04), number 3229 in Lecture Notes in Computer
Science, pages 385–399, Lisbon, Portugal, 2004. Springer Verlag.
[31] Umberto Straccia. Fuzzy description logics with concrete domains. Technical
Report 2005-TR-03, Istituto di Scienza e Tecnologie dell’Informazione,
Consiglio Nazionale delle Ricerche, Pisa, Italy, 2005.
[32] C. Tresp and R. Molitor. A description logic for vague knowledge. In Proc.
of the 13th European Conf. on Artificial Intelligence (ECAI-98), Brighton
(England), August 1998.
[33] John Yen. Generalizing term subsumption languages to fuzzy logic. In
Proceedings of the 12th International Joint Conference on Artificial Intelligence
(IJCAI-91), pages 472–477, Sydney, Australia, 1991.
[34] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.
20
