fuzzyDL: An Expressive Fuzzy Description Logic Reasoner
Fernando Bobillo and Umberto Straccia
Abstract— In this paper we present fuzzyDL, an expressive
fuzzy Description Logic reasoner. We present its salient features,
including some novel concept constructs and queries, and
examples of use cases: matchmaking and fuzzy control.
I. INTRODUCTION
Description Logics (DLs) [2] are a logical reconstruction
of the so-called frame-based knowledge representation lan-
guages, 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]. Indeed, the current standard language
for specifying ontologies is the Web Ontology Language
(OWL) [19], which comprises three sublanguages of increas-
ing expressive power: OWL Lite, OWL DL and OWL Full.
The logical counterpart of OWL Lite and OWL DL are the
DLs SHIF(D) and SHOIN (D), respectively [13].
Fuzzy DLs have been proposed as an extension to clas-
sical DLs with the aim to deal with fuzzy/vague/imprecise
concepts. Since the first work of J. Yen in 1991 [29], an
important number of works can be found in the literature.
For a good survey on fuzzy DLs we refer the reader to [16].
This paper focuses on presenting the fuzzyDL system, an
expressive fuzzy Description Logic reasoner. We present its
salient features and some of its applications.
In the remainder, we proceed as follows. Section II reviews
some preliminaries on fuzzy set theory and fuzzy logic. Sec-
tion III presents the fuzzy DL which is currently supported
by fuzzyDL. Next, Section IV describes its more relevant
features, while Section V presents two use cases (matchmak-
ing and fuzzy control). Finally, Section VI overviews some
related work and presents some conclusions as well as some
ideas for future work.
II. FUZZY SET THEORY AND FUZZY LOGIC
Fuzzy set theory and fuzzy logic were proposed by
Zadeh [30] to manage imprecise and vague knowledge.
While in classical set theory elements either belong to a
set or not, in fuzzy set theory elements can belong to some
degree. More formally, let X be a set of elements called
the reference set. A fuzzy subset A of X, is defined by a
membership function µ
A
(x), or simply A(x), which assigns
any x ∈ X to a value in the real interval between 0 and 1.
Fernando Bobillo is with the Department of Computer Science
and Artificial Intelligence, University of Granada, Spain (email: fbo-
billo@decsai.ugr.es).
Umberto Straccia is with the Istituto di Scienza e Tecnologie
dell’Informazione, Consiglio Nazionale delle Ricerche, Pisa, Italy (email:
straccia@isti.cnr.it).
As in the classical case, 0 means no-membership and 1 full
membership, but now a value between 0 and 1 represents the
extent to which x can be considered as an element of X.
All crisp set operations are extended to fuzzy sets. The in-
tersection, union, complement and implication set operations
are performed by a t-norm function ⊗, a t-conorm function
⊕, a negation function and an implication function ⇒,
respectively. For a definition of these functions we refer the
reader to [14]. Here, we just summarize the fuzzy operators
which will be treated throughout this paper:
Łukasiewicz negation
Ł
α = 1 − α
G
¨
odel t-norm α ⊗
G
β = min{α, β}
Łukasiewicz t-norm α ⊗
Ł
β = max{α + β − 1, 0}
G
¨
odel t-conorm α ⊕
G
β = max{α, β}
Łukasiewicz t-conorm α ⊕
Ł
β = min{α + β, 1}
G
¨
odel implication α ⇒
G
β =
1, if α 6 β
β, if α > β
Łukasiewicz implication α ⇒
Ł
β = min{1, 1 − α + β}
Kleene-Dienes implication α ⇒
Z
β = max{1 − α, β}
where α, β ∈ [0, 1]. Fuzzy implications can be used to define
families of fuzzy operators. We recall here the fuzzy families
that we will consider in this paper:
• Zadeh logic uses Łukasiewicz negation, G
¨
odel t-norm,
G
¨
odel t-conorm and Kleene-Dienes (KD) implication.
• Łukasiewicz logic uses Łukasiewicz negation,
Łukasiewicz t-norm, Łukasiewicz t-conorm and
Łukasiewicz implication.
One of the most important features of fuzzy logic is
its ability to perform approximate reasoning,which involves
inference rules with premises, consequences or both of them
containing fuzzy propositions. A very popular formalism, due
to their practical success, are fuzzy rule-based systems.
A fuzzy IF-THEN system consists of a rule base (a set
of IF-THEN rules) and a reasoning algorithm performing
an inference mechanism such as Rete [7]. In general, the
input of the system is the current value for the input variable
and the output is a fuzzy set, which can be defuzzified into
a single value. In a fuzzy IF-THEN rule, its antecedents,
consequences or both are fuzzy. Fuzzy IF-THEN rules are
fired to a degree which is a function of the degree of match
between their antecedent and the input. The deduction rule is
Generalized Modus Ponens. Roughly speaking, given a rule
“IF A THEN B”, where A and B are fuzzy propositions, it
is possible from a premise “A
0
” which matches A to some
degree, to deduce “B
0
”, which is similar to B.
One of the most popular IF-THEN systems is the Mamdani
model [18]. In a Mamdani model, fuzzy rules have the form
IF X
1
IS A
1
AND . . . AND X
n
IS A
n
THEN Y IS B , (1)
where A
i
and B are linguistic values defined by fuzzy sets
on universes of discourse X
i
and Y respectively, i = 1, . . . n.
For every clause in the antecedent of the rule, the matching
degree between the current value of the variable and the
linguistic label in the rule is computed (typically, using the
minimum or another t-conorm). If there exist several clauses,
they are aggregated into a firing degree, using a fuzzy logic
operator (typically, the maximum). Then, this firing degree
is used for modifying the consequent of the rule using some
function (typically the minimum).
Rules are fired using some inference algorithm. The
computed consequences related to the same variable are
aggregated (typically, using the maximum). Then, the output
variables can be defuzzified. Some examples of defuzzifi-
cation methods are the LOM (largest of maxima), SOM
(smallest of maxima) and MOM (middle of maxima). More
precisely, let B denote the fuzzy set to be defuzzified and
let x be an arbitrary element of the universe. Then for all x:
• x
LOM
is the LOM iff µ
B
(x
LOM
) > µ
B
(x) and, if
µ
B
(x
LOM
) = µ
B
(x) then x
LOM
> x.
• x
SOM
is the SOM iff µ
B
(x
SOM
) > µ
B
(x) and, if
µ
B
(x
SOM
) = µ
B
(x) then x
SOM
< x.
• x
MOM
is the MOM iff x
MOM
= (x
LOM
+ x
SOM
)/2.
III. FUZZY DESCRIPTION LOGICS
We next present the fuzzy DL which is currently behind
the fuzzyDL reasoner, an extension of [23], [24], [25], [26].
However, instead of dealing with the infinite set of truth
degrees in [0, 1] as usual, fuzzyDL restricts to a discretized set
[0, 1]
D
= {0,
1
n
, . . . ,
n−1
n
, 1} for an appropriate big natural
n such that there is a machine representable 0 < <
1
n
.
Assuming a finite set of truth degrees is unavoidable since
fuzzyDL relies on numerical calculus (Mixed Integer Linear
Programming or MILP [21]) using fixed precision, due to
the physical limitations of computers
1
.
A. Syntax
The main ingredients of DLs are concepts, which denote
unary predicates, and roles, which denote binary predicates.
Then there are connectives which allow to construct complex
concepts. For instance, if we use the concept Human to
denote the set of humans, and the concept Male to denote
the set of male objects, the complex concept (conjunction)
Human u Male will denote the set of male humans. On
the other hand, if hasChild denotes a role then the con-
cept Human u ∃hasChild.M ale will denote the the set of
humans having a male child.
fuzzyDL supports concrete data types such as reals, integers,
strings and allows the definition of concepts with explicit
representation of fuzzy membership functions. This is im-
plemented by relying on so-called fuzzy data type theory. A
fuzzy data type theory D = h∆
D
, ·
D
i is such that ·
D
assigns
to every n-ary data type predicate d an n-ary fuzzy relation
over ∆
D
[17]. For instance, the predicate 6
18
may be a
1
It is well known that, e.g., we may loose associativity for very small
numbers, however these cases are very rare in practice.
(a) (b) (c)
(d) (e) (f)
Fig. 1. (a) Trapezoidal function; (b) Triangular function; (c) L-function;
(d) R-function; (e) Crisp interval; (f) Linear function.
unary crisp predicate over the natural numbers denoting the
set of integers smaller or equal to 18. Concerning non-crisp
fuzzy domain predicates, we recall that in fuzzy set theory
and practice, there are many functions for specifying fuzzy
set membership degrees. However, the trapezoidal (Fig. 1
(a)), the triangular (Fig. 1 (b)), the L-function (left-shoulder
function, Fig. 1 (c)), and the R-function (right-shoulder
function, Fig. 1 (d)) are simple, but most frequently used
to specify membership degrees. For backwards compatibil-
ity, fuzzyDL also allows crisp intervals (Fig. 1 (e)). These
functions are defined over the set of non-negative rationals
Q
+
∪ {0} For instance, we may define Y oung : N → [0, 1]
D
to be a fuzzy concrete predicate over the natural numbers
denoting the degree of youngness of a person’s age, as
Y oung(x) = L(10, 30).
fuzzyDL allows fuzzy modifiers, such as very, more or
less and slightly, which apply to fuzzy sets to change
their membership function. Formally, a modifier is a function
f
m
: [0, 1]
D
→ [0, 1]
D
. We will allow modifiers defined in
terms of linear hedges (Figure 1 (f)) and triangular func-
tions (Figure 1 (d)). For instance, very(x) = linear(0.8).
Modifiers have also been considered in [28], [12].
fuzzyDL is based on the fuzzy DL SHIF(D), i.e., SHIF
with concrete data types, which we define next. Let A, R
a
,
R
c
, I, I
c
and M be non-empty finite and pair-wise disjoint
sets of concepts names (denoted A), abstract roles names
(denoted R), concrete roles names (denoted T ), abstract
individual names (denoted x, y), concrete individual names
(denoted v) and modifiers (denoted m). Concepts may be
seen as unary predicates, while roles may be seen as binary
predicates. R
a
also contains a non-empty subset F
a
of
abstract feature names (denoted r), while R
c
contains a
non-empty subset F
c
of concrete feature names (denoted t).
Features are functional roles. The syntax of fuzzy SHIF
concepts (denoted C, D) is as follows:
C, D := > | ⊥ | A | C u D | C t D | ¬C | ∀R.C | ∃R.C
Now, Fuzzy SHIF(D) extends SHIF with concrete
data types, i.e., it has the additional concept constructs:
C, D := ∀T.d | ∃T.d | DR
d := crisp(a, b) | L(a, b) | R(a, b) |
tr iangular(a, b, c) | trapezoidal(a, b, c, d)
D R := > t val | 6 t val | = t val
where val is an integer, a real or a string depending
on the range of the concrete feature t. For instance, the
expression Human u (6 hasAge 18) will denote the set of
humans, which have an age less or equal than 18, while
Human u ∃hasAge.L(10, 30) will denote the set of young
humans (their age is L(10, 30)).
Finally, fuzzyDL extends SHIF(D) as follows:
C, D := C u
G
D | C u
Ł
D | C t
G
D | C t
Ł
D |
C → D | C →
G
D | C →
Ł
D | m(C) |
n C | w
1
C
1
+ · · · + w
k
C
k
| C[> n] | C[6 n]
m := linear(a) | triangular(a, b, c)
where n ∈ [0, 1]
D
, w
i
∈ [0, 1]
D
,
P
k
i=1
w
i
= 1. For instance,
the concept m(C) applies the modifier m to the concept C
and, thus, e.g., Human u ∃hasAge.linear(0.8)(L(10, 30))
denotes the set of very young humans.
A fuzzyDL fuzzy knowledge base (KB) K = hA, T , Ri
consists of a fuzzy ABox A, a fuzzy TBox T and a fuzzy
RBox R.
A fuzzy ABox A consists of a finite set of fuzzy concept
and fuzzy role assertion axioms of the form hx :C, αi and
h(x, y):R, αi, where α ∈ (0, 1]
D
. Informally, from a seman-
tical point of view, hτ, αi constrains the membership degree
of τ to be at least α. Hence, hjim:Y oungP erson, 0.2i says
that jim is a Y oungP erson with degree at least 0.2, while
h(jim, tom) :hasF riend, 1i, states that jim and tom are
friends. If the α is omitted, 1 is assumed.
A fuzzy TBox T is a finite set of fuzzy General Concept
Inclusion axioms (GCIs) hC v D, αi, where α ∈ (0, 1]
D
and C, D are concepts. Informally, hC v D, αi states that all
instances of concept C are instances of concept D to degree
α, that is, the subsumption degree between C and D is at
least α. For instance, hElephant v Animal, 1i states that
the class of elephants is a subclass of the class of animals. We
write C = D as a shorthand of the two axioms hC v D, 1i
and hD v C, 1i. For instance, Minor = P erson u (6
hasAge18) defines a person, whose age is less or equal to
18 (hasAge is a concrete feature), i.e., a minor. If the truth
value α is omitted then the value 1 is assumed.
fuzzyDL also allows to write v
⇒
in order to specify the
particular implication function to be used in the semantics of
the GCI (currently, Kleene-Dienes, Łukasiewicz or G
¨
odel).
Some important particular cases of GCIs are:
• Domain axioms, stating that the domain of role R is the
concept C, i.e., h∃R.> v C, 1i.
• Range axioms, stating that the range of role R is the
concept C, i.e., h> v ∀R.C, 1i.
• Disjointness axioms, stating that concepts C
1
. . . C
k
are
disjoint, i.e., ∀
i,j∈{1,...,k},i<j
, hC
i
u C
j
v ⊥, 1i
A fuzzy RBox R is a finite set of role axioms of the form:
• (fun R), stating that a role R is functional, i.e., R is a
feature.
• (trans R), stating that a role R is transitive.
• R
1
v R
2
, meaning that role R
2
subsumes role R
1
.
• (inv R
1
R
2
), stating that role R
2
is the inverse of R
1
(and vice versa).
A simple role is a role which is neither transitive nor has a
transitive subroles. An important restriction is that functional
needs to be simple.
B. Semantics
The main idea is that concepts and roles are interpreted
as fuzzy subsets of an interpretation’s domain. Therefore,
axioms, rather than being “classical” evaluated (being either
true or false), they are “many-valued” evaluated in [0, 1]
D
.
A fuzzy interpretation I = (∆
I
, ·
I
) relative to a fuzzy data
type theory D = h∆
D
, · i
D
consists of a nonempty set ∆
I
(the domain), disjoint from ∆
D
, and of a fuzzy interpretation
function ·
I
that coincides with ·
D
on every data value, data
type, and fuzzy data type predicate, and it assigns:
• to each abstract concept C a function C
I
: ∆
I
→
[0, 1]
D
;
• to each abstract role R a function R
I
: ∆
I
× ∆
I
→
[0, 1]
D
;
• to each abstract feature r a partial function r
I
: ∆
I
×
∆
I
→ [0, 1]
D
such that for all x ∈ ∆
I
there is an
unique y ∈ ∆
I
on which r
I
(x, y) is defined;
• to each concrete role T a function R
I
: ∆
I
× ∆
D
→
[0, 1]
D
;
• to each concrete feature t a partial function t
I
: ∆
I
×
∆
D
→ [0, 1]
D
such that for all x ∈ ∆
I
there is an
unique v ∈ ∆
D
on which t
I
(x, v) is defined;
• to each modifier m the modifier function f
m
: [0, 1]
D
→
[0, 1]
D
;
• to each abstract individual x an element in ∆
I
;
• to each concrete individual v an element in ∆
D
.
The mapping ·
I
is extended to roles and complex concepts
as specified in Table I, while the mapping ·
I
is extended to
the other constructs as specified in Table II.
The notion of satisfaction of a fuzzy axiom E by a fuzzy
interpretation I, denoted I |= E, is defined as follows:
• I |= hτ > αi iff τ
I
> α,
• I |= (trans R) iff ∀
x,y ∈∆
I
, R
I
(x, y) > sup
z∈∆
I
R
I
(x, z) ⊗ R
I
(z, y),
• I |= R
1
v R
2
iff ∀x, y ∈ ∆
I
.R
1
I
(x, y) 6 R
2
I
(x, y),
• I |= (inv R
1
R
2
) iff ∀x, y ∈ ∆
I
.R
1
I
(x, y) =
R
2
I
(y, x).
We say that concept C is satisfiable iff there is an inter-
pretation I and an individual x ∈ ∆
I
such that C
I
(x) > 0.
For a set of axioms E, we say that I satisfies E iff I
satisfies each element in E. We say that I is a model of E
(resp. E) iff I |= E (resp. I |= E). I satisfies (is a model
of) a fuzzy KB K = hA, T , Ri, denoted I |= K, iff I is a
model of each component A, T and R, respectively.
An axiom E is a logical consequence of a knowledge base
K, denoted K |= E iff every model of K satisfies E.
Given K and a fuzzy axiom τ of the forms hx :C, αi,
h(x, y):R, αi or hC v D, αi, it is of interest to compute τ ’s
best lower degree value bound. The greatest lower bound of
τ w.r.t. K (denoted glb(K, τ)) is glb(K, τ) = sup{n | K |=
hτ > ni}, where sup ∅ = 0. Determining the glb is called
the Best Degree Bound (BDB) problem.
⊥
I
(x) = 0 (m(C))
I
(x) = f
m
(C
I
(x))
>
I
(x) = 1 (∀R.C)
I
(x) = inf
y ∈∆
I
R
I
(x, y) ⇒ C
I
(y)
(¬C)
I
(x) = C
I
(x) (∃R.C)
I
(x) = sup
y ∈∆
I
R
I
(x, y) ⊗ C
I
(y)
(C u D)
I
(x) = C
I
(x) ⊗ D
I
(x) (∀T.d)
I
(x) = inf
y ∈∆
D
T
I
(x, v) ⇒ d
I
(y)
(C u
G
D)
I
(x) = C
I
(x) ⊗
G
D
I
(x) (∃T.d)
I
(x) = sup
y ∈∆
D
T
I
(x, v) ⊗ d
I
(y)
(C u
Ł
D)
I
(x) = C
I
(x) ⊗
Ł
D
I
(x) (n C)
I
(x) = n C
I
(x)
(C t D)
I
(x) = C
I
(x) ⊕ D
I
(x) (w
1
C
1
+ · · · + w
k
C
k
)
I
(x) = w
1
C
1
I
(x) + · · · + w
k
C
k
I
(x)
(C t
G
D)
I
(x) = C
I
(x) ⊕
G
D
I
(x) (C[> n])
I
(x) =
C
I
(x), if C
I
(x) > n
0, otherwise
(C t
Ł
D)
I
(x) = C
I
(x) ⊕
Ł
D
I
(x) (C[6 n])
I
(x) =
C
I
(x), if C
I
(x) 6 n
0, otherwise
(C → D)
I
(x) = C
I
(x) ⇒ D
I
(x) (> t val)
I
(x) = sup
c∈∆
D
t(x, v) ⊗ (v > val)
(C →
G
D)
I
(x) = C
I
(x) ⇒
G
D
I
(x) (6 t val)
I
(x) = sup
c∈∆
D
t(x, v) ⊗ (v 6 val)
(C →
KD
D)
I
(x) = C
I
(x) ⇒
KD
D
I
(x) (= t val)
I
(x) = sup
c∈∆
D
t(x, v) ⊗ (v = val)
(C →
Ł
D)
I
(x) = C
I
(x) ⇒
L
D
I
(x)
TABLE I
SEMANTICS OF THE COMPLEX FUZZY CONCEPTS.
(x:C)
I
= C
I
(x
I
)
((x, y):R)
I
= R
I
(x
I
, y
I
)
(C v D)
I
= inf
x∈∆
I
C
I
(x) ⇒ D
I
(x)
(C v
G
D)
I
= inf
x∈∆
I
C
I
(x) ⇒
G
D
I
(x)
(C v
Ł
D)
I
= inf
x∈∆
I
C
I
(x) ⇒
Ł
D
I
(x)
(C v
KD
D)
I
= inf
x∈∆
I
C
I
(x) ⇒
KD
D
I
(x)
TABLE II
SEMANTICS OF OTHER CONSTRUCTS.
Finally, another similar problem is to compute the best
satisfiability bound of a concept C and amounts to determine
glb(K, C) = sup
I
sup
x∈∆
I
{C
I
(x) | I |= K}. Essentially,
among all models I of the KB, we are determining the
maximal degree of truth that the concept C may have over
all individuals x ∈ ∆
I
.
IV. THE fuzzyDL REASONER
fuzzyDL is a DL reasoner supporting fuzzy logic rea-
soning, publicly available at http://gaia.isti.cnr.
it/
˜
straccia/software/fuzzyDL. It has been de-
veloped in Java, using the parser generator JavaCC
2
and the
MILP-solver Cbc
3
. The reasoning algorithm uses a com-
bination of a tableaux algorithm and a MILP optimization
problem. A detailed description of the reasoning algorithm
cannot fit into this paper, but it can be partially found in [24],
[26], [27]. In the rest of this section, we describe some of
the most interesting features of the reasoner.
a) Expressivity of the representation language: fuzzyDL
extends the classical DL SHIF(D) to the fuzzy case. But
in addition to the constructs of SHIF(D) it allows some
2
https://javacc.dev.java.net
3
http://www.coin-or.org/projects/Cbc.xml
new concepts constructs, already introduced in Section III.
We recall them here since they are not supported by other
fuzzy DL reasoners:
• New concept constructs: weighted concepts (n C),
weighted sum concepts (w
1
C
1
+ · · · + w
k
C
k
) and
threshold concepts [6] (C[> n]) and (C[6 n]).
• Explicit definitions of fuzzy concepts are allowed, by
means of crisp intervals, trapezoidal, triangular, left-
shoulder and right-shoulder membership functions.
• Concept modifiers can be represented using linear
hedges and triangular functions.
• Concrete features or datatypes, which can have a value
with is an integer, a real or a string.
Another interesting feature is that the degrees of the fuzzy
axioms may not only be numerical constants, but also vari-
ables, thus being able to deal with unknown degrees of truth.
b) Fuzzy operators: The user can specify three seman-
tics: Zadeh, Łukasiewicz and classical logics. Classical logic
makes the reasoner backwards compatible, whereas Zadeh
semantics is included for compatibility with most of the
papers about fuzzy DLs. By default, the operators will be
interpreted according to the semantics chosen by the user,
but the language allows to combine arbitrarily the supported
fuzzy operators. Whenever a fuzzy conjunction, a disjunction
or implication is used, the user can specify its particular
semantics.
The following restriction should be remarked. If G
¨
odel
implication ⇒
G
appears in the fuzzy KB, then the reasoner
restricts the calculus to finite models. In general, G
¨
odel logic
does not have the witnessed model property, i.e., there can
exist fuzzy KBs which have an infinite model, but they do not
have a witnessed model (see [11] for an example). However,
due to the limited precision of computers, we will deal with a
finite number of truth values, and in G
¨
odel logic over a fixed
finite subset of truth values all models (finite or infinite) are
witnessed [11].
c) Constraints: The possible values of the variables in
the KB may be restricted by specifying linear inequations, as
well as by restricting variables to be binary or free. Clearly,
the range of the variables can be defined by means of a pair
of linear inequations. As a consequence, it is possible to
represent (and to solve) MILP problems using fuzzyDL. We
allow the following constraints, with ∈ {>, 6, =}:
Constraint Semantics
(linear inequation) a
1
var
1
+ · · · + a
k
∗ var
k
n
(binary var) var ∈ {0, 1}
(free var) var ∈ (−∞, ∞)
For instance, we may encode that “Tim is taller than Tom”
by specifying htim:T all, x
1
i, htom:¬T all, x
2
i with the
constraint x
1
+ x
2
= 1.
d) Query language: fuzzyDL allows to compute several
different types of queries. It does not only allow typical
reasoning tasks such as the BDB, concept satisfiability and
subsumption problems, but it also allows to optimize vari-
ables and to perform defuzzifications. The concrete syntax
for the queries is the following:
Query Semantics
(max-instance? a C) sup{n | K |= ha : C, n i}
(min-instance? a C) inf{n | K |= ha : C, ni}
(max-related? a b R) sup{n | Kh(a, b) : R, ni}
(min-related? a b R) inf{n | K |= h(a, b) : R, ni}
(max-subs? C D) sup{n | K |= hD v C, ni}
(min-subs? C D) inf{n | K |= hD v C, ni}
(g-max-subs? C D) sup{n | K |= hD v
G
C, ni}
(g-min-subs? C D) inf{n | K |= hD v
G
C, ni}
(l-max-subs? C D) sup{n | K |= hD v
Ł
C, ni}
(l-min-subs? C D) inf{n | K |= hD v
Ł
C, ni}
(kd-max-subs? C D) sup{n | K |= hD v
KD
C, ni}
(kd-min-subs? C D) inf{n | K |= hD v
KD
C, ni}
(max-sat? C [a]) sup
I
sup
a∈∆
I
C
I
(a)
(min-sat? C [a]) inf
I
sup
a∈∆
I
C
I
(a)
(max-var? var) sup{var | K is consistent}
(min-var? var) inf{var | K is consistent}
(defuzzify-lom? C a t) defuzzify t using LOM
(defuzzify-som? C a t) defuzzify t using SOM
(defuzzify-mom? C a t) defuzzify t using MOM
It is worth to note that in concept satisfiability queries
(max-sat? and min-sat?) it is possible the use an individual
a instead of an arbitrary one. This will be used, for example,
in the computation of the defuzzification methods, which will
be explained in Section V-B together with an example.
e) Output: In most of the applications, the user is not
only interested in the answer of the query, but also in the
fuzzy model which satisfies it. However, not all elements
of the model are relevant to the user, specially if he is
working with large fuzzy KBs. fuzzyDL provides several
statements which allow the user to restrict the elements of the
fuzzy model for which the value will be shown. The current
available statements allow to show:
• the value of the fillers of the concrete roles R
1
. . . R
n
for every individual,
• the value of the fillers of the concrete roles R
1
. . . R
n
for a particular individual a,
• the value of the variables x
1
. . . x
n
,
• the value of the instances of the atomic concepts
A
1
. . . A
n
,
• the membership degree of the individuals a
1
. . . a
n
to
any atomic concept,
• the language of the KB, ranging from ALC to
SHIF(D).
f) Import of OWL ontologies: At this moment, there
does not exist an editor allowing to create fuzzy KBs in
fuzzyDL. Up to now, the user needs to create a text file using
the syntax of the language.
In order to make the representation of fuzzy KBs easier,
fuzzyDL also allows the possibility of importing OWL Lite
ontologies. These (crisp) ontologies are saved into a text file
which the user can edit and extend, for example adding
membership degrees to the fuzzy axioms or specifying a
particular fuzzy operator for a complex concept.
V. SOME EXAMPLES OF USE CASES
A. Matchmaking
Assume, that a car seller sells a sedan car. A buyer is looking
for a second hand passenger car. Both the buyer as well as the
seller have preferences (restrictions). Our aim is to find the
best agreement. The preferences are as follows. Concerning
the buyer:
1) He does not want to pay more than 26000 euro (buyer
reservation value).
2) If there is an alarm system in the car then he is
completely satisfied with paying no more than 22300
euro, but he can go up to 22750 euro to a lesser degree
of satisfaction.
3) He wants a driver insurance and either a theft insurance
or a fire insurance.
4) He wants air conditioning and the external colour
should be either black or grey.
5) Preferably the price is no more than 22000 euro, but
he can go up to 24000 euro to a lesser degree of
satisfaction.
6) The kilometer warranty is preferrably at least 175000,
but he may go down to 150000 to a lesser degree of
satisfaction.
