REASONING WITH THE FUZZY DESCRIPTION LOGIC f
Z
SI
Jidi Zhao
1
, Harold Boley
2
and Weichang Du
1
1
Faculty of Computer Science, University of New Brunswick, 540 Windsor Street, Fredericton, Canada
2
Institute for Information Technology, National Research Council of Canada, 46 Dineen Drive, Fredericton, Canada
Keywords:
Semantic web, Uncertain knowledge, Description logic, Fuzzy logic, Linear programming.
Abstract:
While applications in different areas have shown the necessity of dealing with uncertain knowledge, Semantic
Web techniques based on standard Description Logics do not have such a capability. Motivated by this discrep-
ancy, we introduce an expressive fuzzy description logic, f
Z
SI , which extends the classic Description Logic
SI to deal with uncertain knowledge about concepts and roles as well as instances of concepts and roles. In the
family of Fuzzy Logics it is semantically based on Zadeh Logic, which naturally interprets uncertain knowl-
edge about concepts and roles as fuzzy sets and fuzzy relations, and interprets uncertain knowledge about
instances as elements with degrees of membership. The paper focuses on several reasoning methods for the
main reasoning problems in f
Z
SI , including consistency checking, instance range entailment, and f-retrieval
problems.
1 INTRODUCTION
The Semantic Web initiative aims at creating an ex-
tension to the current World Wide Web by develop-
ing logic-based standards and technologies that en-
able machines to understand the information on the
Web, so that they can support richer knowledge infer-
ence and automate the performance of various tasks
for human beings (Berners-Lee et al., 2001).
The current W3C standard for Semantic Web on-
tology languages, Web Ontology Language (OWL),
is designed for use by applications that need to pro-
cess the content of information instead of just pre-
senting information to humans (McGuinness and van
Harmelen, 2004). It facilitates greater machine in-
terpretability of Web content than that supported by
other Web languages such as XML, RDF, and RDF
Schema (RDFS). This ability of OWL is enabled by
its underlying knowledge representation formalism of
Description Logics (DLs). DLs (Baader et al., 2003;
Horrocks and Sattler, 1999) are a family of logic-
based formalisms designed to represent and reason
about the conceptual knowledge of arbitrary domains.
Elementary descriptions of DLs are atomic concepts
and atomic roles. Complex concept descriptions and
role descriptions can be built from the elementary de-
scriptions according to construction rules. Different
description languages of DLs are distinguished by the
kind of concept and role constructors (such as conjun-
ction, disjunction, and exists restriction) allowed in
their description language and the kinds of axioms al-
lowed in their terminologies. The basic proposition-
ally closed DL is ALC in which the letters AL stand
for attributive language and the letter C for comple-
ment (negation of arbitrary concepts). Besides ALC,
other letters are used to indicate various DL exten-
sions. For example, in the Description Logic S I (Hor-
rocks and Sattler, 1999), S is used for ALC extended
with transitive roles (R
+
), and I for inverse roles. DLs
have a model-theoretic semantics, which is defined by
interpreting concepts as sets of individuals and roles
as sets of pairs of individuals. An interpretation I is
a pair I = (
I
,·
I
) consisting of a domain
I
which is
a non empty set and of an interpretation function ·
I
which maps each individual x into an element of
I
(x
I
), each concept C into a subset of
I
(C
I
I
)
and each role R into a subset of
I
×
I
(R
I
I
×
I
).
The semantics of complex concept and role descrip-
tions can be found in (Baader et al., 2003; Horrocks
and Sattler, 1999). Furthermore, a knowledge base
(KB) in DLs consists of two parts: the terminological
box (TBox T ) and the assertional box (ABox A).
Uncertainty is an intrinsic feature of real-world
knowledge, which is also reflected in the World Wide
Web and the Semantic Web. Many concepts needed in
knowledge modeling lack well-defined boundaries or,
precisely defined criteria. Examples are the concepts
of young, tall, and cold. The Uncertainty Reason-
21
Zhao J., Boley H. and Du W..
REASONING WITH THE FUZZY DESCRIPTION LOGIC fZSI.
DOI: 10.5220/0003054700210030
In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages
21-30
ISBN: 978-989-8425-32-4
Copyright
c
2010 SCITEPRESS (Science and Technology Publications, Lda.)
ing for the World Wide Web (URW3) Incubator Group
defined the challenge of representing and reasoning
with uncertain information on the Web. According
to the latest URW3 draft report, uncertainty is a term
intended to encompass different forms of uncertain
knowledge, including incompleteness, inconclusive-
ness, vagueness, ambiguity, and others (Laskey et al.,
2008). The need to model and reason with uncer-
tainty has been found in many different Semantic
Web contexts, such as matchmaking in Web services
(Martin-Recuerda and Robertson, 2005), classifica-
tion of genes in bioinformatics (Stevens et al., 2007),
multimedia annotation (Stamou et al., 2006), and on-
tology learning (Haase and V
¨
olker, 2005).
Fuzzy Set Theory was first introduced by Zadeh
(Zadeh, 1965) as an extension to the classic notion of
a set to capture inherent vagueness (the lack of crisp
boundaries of sets). Fuzzy Logic is a form of multi-
valued logic derived from Fuzzy Set Theory to deal
with reasoning that is approximate rather than precise.
In Fuzzy Logic, the degree of truth of a statement can
range between 0 and 1, and is not constrained to the
two truth values {0,1} or { f alse,true} as in classic
predicate logic. Formally, a fuzzy set A with respect
to a set of elements (also called a universe) is char-
acterized by a membership function µ
A
(x) which as-
signs a value in the real unit interval [0,1] to each ele-
ment x in (x ), notated as µ
A
: [0,1]. µ
A
(x),
often written as A(x), gives the degree of an element x
belonging to the set A. Such degrees can be computed
based on a membership function. A fuzzy relation R
over two fuzzy sets A and B is similarly defined by a
function R : × [0, 1].
Fuzzy Logic extends the Boolean operations de-
fined on crisp sets and relations for fuzzy sets and
fuzzy relations. These operations, e.g. complement,
union, and intersection, are interpreted as mathemat-
ical functions over the unit interval [0,1]. In the fol-
lowing, η,θ define the truth degrees of sets and re-
lations, ranging between 0 and 1. The mathemati-
cal functions for fuzzy intersection are usually called
t-norms (t(η,θ)); those for fuzzy union are called
s-norms (s(η,θ), a.k.a. t-conorms); and those for
the fuzzy set complement are called negations (¬η);
These functions usually satisfy certain mathematical
properties. The most widely known operations in the
Fuzzy Logic family are Zadeh Logic, Lukasiewicz
Logic, Product Logic, and G
¨
odel Logic.
To deal with the ‘crisp limitation’ of classic DLs,
considerable work has been carried out on integrating
uncertain knowledge into DLs in recent years. The
current literature generally follows two approaches.
One is Probabilistic Logic based on Probability The-
ory; for example the work in (Jaeger, 1994; Koller
et al., 1997; Lukasiewicz, 2008). The other is Fuzzy
Logic and Fuzzy Sets; for example the work in (Yen,
1991; Straccia, 2001; Zhao and Boley, 2010). A re-
view and comparison of these works can be found in
(Zhao, 2010). We presented a Norm-Parameterized
Fuzzy Description Logic fALC N and addressed the
consistency checking problem in (Zhao and Boley,
2010). We use f
N
DL to denote a Fuzzy Descrip-
tion Logic fDL with norm parameter N. Omitting
the index N means the fDL is norm-parameterized.
In the current paper, we follow the Fuzzy Sets and
Fuzzy Logic approach and present the fuzzy Descrip-
tion Logic f
Z
SI . We call this fuzzy Description Logic
f
Z
SI as S I is the underlying Description Logic and Z
fixes the norms to Zadeh Logic. This paper is differ-
ent from previous work due to the following features.
First, the underlying classic DL S I is a more expres-
sive Description Logic which deals with fuzzy transi-
tive roles and fuzzy inverse roles. Second, we com-
bine Description Logic, Fuzzy Logic, and Linear Pro-
gramming methods in the reasoning procedure. Last
but not least, f
Z
SI supports both fuzzy axioms and
fuzzy assertions for uncertain knowledge representa-
tion and reasoning.
2 THE FUZZY DL f
Z
SI
f
Z
SI extends the f
Z
ALC N DL with inverse roles,
and transitive roles but excludes number restrictions.
Due to space limitations, we refer interested readers
to (Zhao and Boley, 2010) for the syntax and seman-
tics of complex concept descriptions as well as ax-
ioms and assertions for f
Z
ALC by specializing the
t-norm to min and the s-norm to max. Here we sim-
ply list them in Tables 1 and 2, and then explain the
expressiveness beyond f
Z
ALC . A fuzzy knowledge
base in f
Z
SI consists of two parts: the fuzzy termino-
logical box consisting of a finite set of fuzzy axioms
(TBox T ) and the fuzzy assertional box consisting of
a finite set of fuzzy assertions (ABox A). As shown
in Table 2, a fuzzy axiom or fuzzy assertion is of the
form α [l,u] with 0 l u 1, which is equivalent to
the two inequalities α l and α u. In what follows,
we use these expressions as needed.
In classic DLs, a role R is symmetric iff for all
x,y 4
I
, (Inv(R))
I
(y,x) = R
I
(x,y), where the role
function Inv(R) defines the inverse of a role. The
same property holds for a fuzzy symmetric role. For
example, the role hasPart is the inverse of the role
isPartOf.
In classic DLs, a role R is transitive iff for all
x,y,z 4
I
, R
I
(x,y) and R
I
(y,z) imply R
I
(x,z). While
in Fuzzy Logic, a fuzzy role R is transitive iff for all
ICFC 2010 - International Conference on Fuzzy Computation
22
Table 1: Syntax and semantics of f
Z
SI constructors.
Constructor Syntax Semantics
top concept > >
I
= 1
bottom
concept
I
= 0
atomic
negation
¬A (¬A)
I
(x) = 1 A
I
(x)
concept
conjunction
C uD (C u D)
I
= min(C
I
(x),D
I
(x))
concept
disjunction
C tD (C t D)
I
= max(C
I
(x),D
I
(x))
exists
restriction
R.C (R.C)
I
(x) =
sup
y
I
{min(R
I
(x,y),C
I
(y))}
value
restriction
R.C (R.C)
I
(x) = inf
y
I
{max(1
R
I
(x,y),C
I
(y))}
inverse role Inv(R) (Inv(R))
I
(y,x) = R
I
(x,y)
Table 2: Syntax and semantics of f
Z
SI axioms.
Axioms Syntax Semantics
concept
inclusion
A v C x
I
,A
I
(x) C
I
(x)
concept
definition
A C x
I
,A
I
(x) = C
I
(x)
concept
implica-
tion
A
C [l,u]
x
I
,C
I
(x)
min(A
I
(x),[l,u])
transitive
role
Trans(R) R
I
(a,c)
sup
b∈4
I
min(R
I
(a,b),R
I
(b,c))
concept
assertion
C(a) [l,u] l C
I
(a) u
role asser-
tion
R(a,b) [l, u] l R
I
(a,b) u
individual
inequality
a 6= b a
I
6= b
I
x,y,z 4
I
, it satisfies the following inequality (D
´
ıaz
et al., 2010):
R
I
(x,z) sup
y∈4
I
t(R
I
(x,y),R
I
(y,z)) (1)
where t(η,θ) denotes a general t-norm. Thus, in the
case of Zadeh Logic, a transitive role satisfies:
R
I
(x,z) sup
y∈4
I
min(R
I
(x,y),R
I
(y,z)) (2)
In order to make the following explanations eas-
ier, we introduce the role function Trans(R) which
specifies that R is transitive or Inv(R) is transitive.
Now, we use some mathematical properties of
Zadeh Logic to show that the following property
is satisfied by a role value restriction R.C with
Trans(R).
Lemma 1. Under Zadeh Logic, if (R.C)
I
(x) l (l
[0,1]) and R is transitive, then (R.(R.C))
I
(x) l
holds.
Proof. (R.C)
I
(x) l
De f inition o f semantics
inf
z
I
{max(¬R
I
(x,z),C
I
(z))} l
Equation 1
inf
z
I
inf
y
I
{max(¬(min(R
I
(x,y),R
I
(y,z))),C
I
(z))} l
De Morgan
0
s Law
inf
z
I
inf
y
I
{max(max(¬R
I
(x,y),¬R
I
(y,z)),C
I
(z))} l
Associativity
inf
z
I
inf
y
I
{max(¬R
I
(x,y),max(¬R
I
(y,z),C
I
(z)))} l
Commutativity
inf
y
I
{max(¬R
I
(x,y),inf
z
I
max(¬R
I
(y,z),C
I
(z)))} l
De f inition o f semantics
inf
y
I
{max(¬R
I
(x,y),(R.C)
I
(y))} l
De f inition o f semantics
(R.(R.C))
I
(x) l
However, in the cases of , we cannot derive such
a property for (R.C)(x) and Trans(R).
Under Zadeh Logic, by applying the semantics of
R.C and negation, it is easy to see that the following
equivalence rules hold:
a,b
I
,
¬¬C C , (3)
¬∃R.C R.¬C , (4)
¬∀R.C R.¬C (5)
Then, (R.C)
I
(x) u
Monotonicity
¬((R.C)
I
(x)) 1 u
Equilvalence 4
(R.(¬C))
I
(x)) 1 u
Lemma 1
(R.(R.(¬C)))
I
(x)) 1 u
Monotonicity
¬(R.(R.(¬C))
I
(x)) u
Equilvalence 5
(R.¬(R.(¬C)))
I
(x) u
Equilvalence 5 and 3
(R.(R.C))
I
(x) u
Therefore, the following property is satisfied with
respect to a role exists restriction R.C and Trans(R).
Such a property cannot be inferred from the cases of
.
Lemma 2. Under Zadeh Logic, if (R.C)
I
(x) u and
R is transitive, then (R.(R.C))
I
(x) u holds.
Although we can show that such properties also
hold under Product Logic and other logics, we ne-
glect it here, as it is out of scope. We will soon see
that these properties will be embodied in the fuzzy
completion rules for the f
Z
SI reasoning algorithm.
REASONING WITH THE FUZZY DESCRIPTION LOGIC fZSI
23
3 REASONING ALGORITHM
FOR BUILDING A FUZZY
TABLEAU OF f
Z
SI
The reasoning algorithm that we will present is a
fuzzy extension to the tableau method and tests the
consistency of a knowledge base KB =< T , A > by
trying to construct a model of KB. A model of KB
in our Fuzzy Description Logic f
Z
SI is a fuzzy in-
terpretation I = (4
I
,·
I
). Similar to the classic DL,
such a model has the shape of a forest, i.e., a col-
lection of trees, with nodes corresponding to indi-
viduals, root nodes corresponding to named individ-
uals, and edges corresponding to roles between indi-
viduals. Each node has a node label L(individual),
but different from classic DLs, each node in a f
Z
SI
tableau is labeled with a set of f
Z
SI -concepts. Each
element in the set consists of a pair of elements
{concept,constraint}. The sets for all nodes are re-
stricted to subsets of sub(A), where sub(A) is the
set of sub-concepts of concepts that appear within an
ABox A. Furthermore, each edge is associated with
an edge label L(individual
1
, individual
2
) which con-
sists of a pair of elements {role,constraint}.
In (Zhao and Boley, 2010), we explained the TBox
processing procedure which consists of some prepro-
cessing steps to deal with the fuzzy TBox before ap-
plying the reasoning algorithm. Those steps are appli-
cable to the f
Z
SI knowledge base. Therefore, we can
assume all concepts C occurring in KB to be in nega-
tion normal form (NNF) and we only deal with un-
foldable TBox after those preprocessing steps. How-
ever, due to the properties of a f
Z
SI knowledge base,
the TBox processing procedure should include a cou-
ple of other steps. First, the TBox processing proce-
dure transforms all the assertions in the fuzzy ABox
and the fuzzy implication axioms in the fuzzy TBox
with the form α
0
[l, u] into two expressions: α
0
l
and α
0
u. In order to keep our presentation simple
and compact, in what follows, we use a general form
α op n where op {≥, ≤} and n [0,1] whenever
applicable. Second, an f
Z
SI knowledge base may
contain transitive role axioms and inverse roles. We
know that if a role R is transitive, the inverse role of R
is also transitive. Therefore, for each pair of Trans(R)
and Inv(R), the procedure should also add an axiom
Trans(Inv(R)). After the application of the TBox pro-
cessing procedure, in what follows, we only have to
consider a knowledge base in f
Z
SI only consists of
fuzzy ABox assertions, a set of transitive role axioms,
and a finite set of fuzzy implication axioms.
Next, we first present the definitions of fuzzy
tableau, clash, and clash-free, and then prove the rela-
tion between the consistency of a fuzzy knowledge
base KB =< T ,A > and the existence of a fuzzy
tableau T for KB.
Definition 1. If KB =< T ,A > is an f
Z
SI knowledge
base, R
A
is the set of roles occurring in A, together
with their Inv(R)s, a fuzzy tableau T for KB is defined
to be a quadruple (S, L, ε,C ) such that: S is a set of
individuals, L : S × sub(A) [0,1] maps each indi-
vidual and a concept which is a subset of sub(A) to
the membership degree of the individual to that con-
cept, ε : R
A
×S ×S [0, 1] maps each role in R
A
and
a pair of individuals to the membership degree of the
pair to the role, and C is a set of constraints must be
satisfied. For all x,y S, A,C,D sub(A), R R
A
and n [0, 1], it holds that:
1. For any x S, {x : = 0} and {x : > = 1} L(x).
2. If {x : ¬(A) op n} L(x), then {x : A op 1 n}
L(x).
3. If {x : C uD n} L(x), then {x : C n} L(x)
and {x : D n} L(x).
4. If {x : C tD n} L(x), then {x : C n} L(x)
and {x : D n} L(x).
5. If {x : C u D n} L(x), then {x : C n
1
}
L(x), {x : D n
2
} L(x), and n = min(n
1
,n
2
)
for some n
1
,n
2
.
6. If {x : C t D n} L(x), then {x : C n
1
}
L(x), {x : D n
2
} L(x), and n = max(n
1
,n
2
)
for some n
1
,n
2
.
7. If {x : R.C n} L(x), then there exists y S
such that {< x, y >: R n} ε(R) and {y : C
n} L(y).
8. If {x : R.C n} L(x), then there exists y S
such that {< x, y >: R 1 n} ε(R) and {y :
C n} L(y).
9. If {x : R.C n} L(x), then {< x,y >: R
n
1
} ε(R), {y : C n
2
} L(y), and n =
min(n
1
,n
2
) for some n
1
,n
2
.
10. If {x : R.C n} L(x), then {< x,y >: R 1
n
1
} ε(R), {y : C n
2
} L(y), and n = max(1
n
1
,n
2
) for some n
1
,n
2
.
11. {< x, y >: R op n} ε(R) iff {< y,x >:
Inv(R) op n} ε(R).
12. If {x : R.C n} L(x) and Trans(R), then
{< x,y >: R 1 n
1
} ε(R), {y : R.C n
2
}
L(y), and n = max(1 n
1
,n
2
) for some n
1
,n
2
.
13. If {x : R.C n} L(x) and Trans(R), then {<
x,y >: R n
1
} ε(R), {y : R.C n
2
} L(y),
and n = min(n
1
,n
2
) for some n
1
,n
2
.
14. If {A C n} T and {x : A n
1
} L(x),
then {x : C n
2
} L(x) and n
2
= min(n,n
1
),
for any x S.
ICFC 2010 - International Conference on Fuzzy Computation
24
15. If {A C n} T and {x : A n
1
} L(x),
then {x : C n
2
} L(x) and n
2
= min(n,n
1
),
for any x S.
In (Zhao and Boley, 2010), we defined the se-
mantics (C u D)
I
as t(C
I
(x),D
I
(x)) for various t-
norms. For the case of Zadeh Logic, we have that
if (C u D)
I
(x) n, then C
I
(x) = n
C
, D
I
(x) = n
D
, and
min(n
C
,n
D
) n. In this definition, we can draw a
further conclusion based on the properties of the min
norm that C
I
(x) = n
C
n and D
I
(x) = n
D
n. Sim-
ilar extensions are conducted on other f
Z
SI concepts
and roles.
Definition 2. Let A be an extended f
Z
SI ABox, A
contains a clash if only if one of the following situa-
tions occurs:
1. {⊥(a) 6= 0} A
2. {>(a) 6= 1} A
3. {α n
1
,α n
2
} A and n
1
< n
2
4. there is no solution for the constraint system of
inequations C
A is called clash-free if it does not contain any clash.
Lemma 3. An f
Z
SI knowledge base KB =< T ,A >
is consistent iff there exists a clash-free fuzzy tableau
for KB.
Proof. For the if direction, if T=(S,L,ε) is a clash-
free fuzzy tableau for a fuzzy knowledge base KB, a
fuzzy interpretation I=(
I
,·
I
) can be constructed as:
I
= S
>
I
={x : > = 1} L(x) for any x in S
I
={x : = 0} L(x) for any x in S
A
I
={x : A op n} L(x) for all concept names A in
sub(A)
R
I
=
(
ε(R)
+
i f Trans(R)
ε(R) otherwise
where ε(R)
+
denotes the fuzzy sup-min transitive
closure of ε(R) (Lee, 2001; Mitsuishi and Bancerek,
2003).
To prove that I is a model of KB, we show by
induction on the structure of concepts that, if {x :
E op n} L(x), then E
I
(x) op n for any x in S. With-
out loss of generality, we only show in the following
the cases with {x : E n} L(x).
1. If E the > or concept, and {x : = 0} and
{x : > = 1} L(x), then by definition, >
I
(x) = 1
or
I
(x) = 0.
2. If E is a concept name other than > and , and
{x : E n} L(x), then E
I
(x) n by definition.
3. If E = ¬(C) and {x : ¬(C) n} L(x), then {x :
C 1n} L(x) (due to Property 2 in Definition
1), so we have C
I
(x) 1 n by induction. Hence,
(¬C)
I
(x) 1 (1 n) = n, i.e., E
I
(x) n.
4. If E = (C
1
uC
2
) and {x : C
1
uC
2
n} L(x), then
{x : C
1
n} L(x) and {x : C
2
n} L(x), so by
induction (C
1
)
I
(x) n and (C
2
)
I
(x) n. Hence,
(C
1
uC
2
)
I
(x) = min((C
1
)
I
(x),(C
2
)
I
(x)) n.
5. If E = (C
1
t C
2
) and {x : C
1
t C
2
n} L(x),
since the tableau is clash free, we can find some
n
1
,n
2
so that {x : C
1
n
1
} L(x), {x : C
2
n
2
} L(x) and n = max(n
1
,n
2
). By induc-
tion (C
1
)
I
(x) n
1
, (C
2
)
I
(x) n
2
. Hence, (C
1
u
C
2
)
I
(x) = max((C
1
)
I
(x),(C
2
)
I
(x)) n.
6. If E = (S.C) and {x : S.C n} L(x), then
there exists y S such that {< x,y >: S n}
ε(S) and {y : C n} L(y), so by induction
S
I
(x,y) n and C
I
(y) n. Hence (S.C)
I
(x) =
sup
y
I
min(S
I
(x,y),C
I
(y)) n.
7. If E = (S.C), {x : S.C n} L(x), and
S
I
(x,y) = m, then it would be either of the fol-
lowing two cases.
{< x, y >: S = m} ε(S): if m > 1 n, then
{y : C n} L(y) (due to Property 10 in Def-
inition 1), so we have S
I
(x,y) > 1 n and
C
I
(y) n, hence, (S.C)
I
(x) = in f
y
I
max(1
S
I
(x,y),C
I
(y)) n; if m <= 1 n, then 1
S
I
(x,y) = 1 m n, hence (S.C)
I
(x) =
in f
y
I
max(1 S
I
(x,y),C
I
(y)) n.
{< x, y >: S = m} / ε(S) and there exist l paths
(l 1) such that in each path, {< x,x
l1
>: S =
m
l1
} ε(S), {< x
l1
,x
l2
>: S = m
l2
} ε(S), · ··,
{< x
ln
,y >: S = m
l(n+1)
} ε(S) and Trans(R).
Thus, the truth degree of < x,y > to the tran-
sitive closure of S, m, would be equal to the
supremum value among all the minimum val-
ues of each path. In this case: if m > 1 n,
then there exists at least one path k, {< x, x
k1
>:
S = m
k1
} ε(S), {< x
k1
,x
k2
>: S = m
k2
}
ε(S), ·· ·, {< x
kn
,y >: S = m
k(n+1)
} ε(S), we
have m
ki
> m > 1 n (1 i (n + 1)) (as
m is the minimum value of the path), {x
ki
:
(S.C) n} L(x
ki
) (1 i n), and {y :
(S.C) n} L(y) (due to Property 12 in Def-
inition 1), so, inducted from {< x, x
k1
>: S >
1 n} ε(S) and {x
k1
: (S.C) n} L(x
k1
),
we have S
I
(x,x
k1
) > 1 n and (S.C)
I
(x
k1
)
n, and thus C
I
(x
k1
) n, hence (S.C)
I
(x) =
in f
y
I
max(1 S
I
(x,x
k1
),C
I
(x
k1
)) n; if
m <= 1 n, then we have max(1m,C
I
(y))
n, hence (S.C)
I
(x) n.
The cases for the inequalities can be proved in
a similar way.
For the converse, if I=(
I
,·
I
) is a model of A, then
a fuzzy tableau T=(S,L,ε) can be defined as:
S =
I
REASONING WITH THE FUZZY DESCRIPTION LOGIC fZSI
25
ε(R) = R
I
L(x) = {x : C op n} for all x S and C sub(A)
To prove that T is a fuzzy tableau of KB, we show
that, based on I, all the properties in Definition 1 are
satisfied.
1. T satisfies Property 1 - 12, 14, and 15 as a di-
rect consequence of the semantics of f
Z
SI con-
cepts. For example, let {x : C u D n} L(x),
the semantics of fuzzy concept conjunction im-
plies that (C u D)
I
(x) = min(C
I
(x),D
I
(x)) n,
thus we have D
I
(x) n and D
I
(x) n, that is,
{x : C n} L(x) and {x : D n} L(x), hence
Property 3 is satisfied. For similar reasons, other
properties hold.
2. Property 12 of Definition 1 is satisfied as a re-
sult of the semantics and the properties of tran-
sitive roles and value restrictions that have been
investigated in Section 2. Hence, if (R.C)
I
(x)
n, Trans(R) then (R.(R.C))
I
(x) n, thus
R
I
(s,t) 1 n
1
, (R.C)
I
(t) n
2
and n =
max(1 n
1
,n
2
) hold.
3. Similarly, Property 13 is satisfied as a result of
the semantics and the properties of transitive roles
and role exists restrictions.
From Lemma 3, an algorithm that constructs a
fuzzy tableau for an f
Z
SI knowledge base can be used
as a decision procedure for the consistency checking
problem.
Similar to the tableau algorithm presented by Hor-
rocks et al. (Horrocks et al., 2000), our algorithm
works on building a fuzzy tableau for an f
Z
SI knowl-
edge base which may be a completion-forest since
the ABox might contain several named individuals
with arbitrary edges connecting them. Each node x
is labeled with a set L(x) = {{x : C
1
op n
1
},·· · ,{x :
C
m
op n
m
}} (m 1) and a constraint set C (x) =
{{x
C
1
op n
1
},·,{x
C
m
op n
m
}}, where C
i
sub(A),
x
C
i
,n
i
[0,1], 1 i m, and op {≥,≤}. Each edge
< x,y > is labeled with a set L(x,y) = {[x,y] : R op n}
and a constraint in the set C (x, y) = {x
R
op n}, where
R are roles occurring in A.
We adapt the conjugation concept in (Straccia,
2001) to represent pairs of fuzzy assertions that form
a contradiction. Let α be a SI assertion, two fuzzy as-
sertions (α n
1
and α n
2
) conjugate with each
other if n
1
> n
2
. For a given fuzzy assertion, its conju-
gated assertion is not unique, and in fact, infinite. For
example, both {[x,y] : R 0.5} and {[x,y] : R 0.3}
conjugate with the fuzzy assertion {[x,y] : R 0.6}.
Let us recall some notations used in (Horrocks
and Sattler, 1999). If nodes x and y are connected
by an edge < x,y > with {R op n} L(x,y), then y
is called an R
n
-successor of x and x is called an R
n
-
predecessor of y. Ancestor is the transitive closure of
predecessor. If y is an R
n
-successor or an (Inv(R))
n
-
predecessor of x, then y is called an R
n
-neighbor of
x. An expressive DL such as f
Z
SI which allows tran-
sitive roles and inverse roles may lead to nontermina-
tion as the fuzzy completion rules can introduce new
concepts that are the same size as the decomposed
concept. Our algorithm for the consistency check-
ing of an f
Z
SI knowledge base follows the dynamic
blocking presented in (Horrocks and Sattler, 1999)
and uses it to guarantee the termination of the reason-
ing algorithm. In dynamic blocking, blocked nodes
are allowed to be dynamically established and broken
as the expansion progresses, and expand role value re-
striction and role exists restriction concepts. This dy-
namic blocking strategy is crucial in the presence of
inverse roles since information might be propagated
up the completion-forest and affect other branches.
For example, consider the nodes x, y and z, the edges
< x,y > and < x, z >. Suppose x blocks y. In the
presence of inverse roles it is possible that z adds in-
formation to node x, although z is a successor of x.
In that case the block on y must be broken. A node
x is blocked if for some ancestor y, y is blocked or
L(x) = L(y). Dynamic blocking uses the notions of
directly blocked and indirectly blocked nodes. If a
blocked node xs predecessor is blocked, x is called
indirectly blocked. A blocked node x is called di-
rectly blocked if it has a unique ancestor y such that
L(x) = L(y).
Now, for an expanded f
Z
SI ABox A with a set of
transitive role axioms and a set of fuzzy implication
axioms, the algorithm initializes a forest to contain
(1)root nodes, for each individual x occurring in A,
the root node x is labeled with L(x) = {x : C op n}
and C (x) = {x
C
op n} for each assertion of the form
C(x) op n in A, and (2)edges, each edge hx, yi cor-
responds to an assertion R(x,y) op n in A with R be
an atomic role or an inverse role and is labeled with
L(x, y) = {[x,y] : R op n} and C (x,y) = {x
R
op n}. If
an assertion is of the form Inv(P)(x,y) op n, the cor-
responding edge is also labeled with L(x,y) = {[y, x] :
P op n} and C(x,y) = {x
P
op n}. The completion for-
est is then expanded by repeatedly applying the fol-
lowing fuzzy completion rules in Table 3. The com-
pletion forest is complete when a clash is detected, or
none of the fuzzy completion rules are applicable.
ICFC 2010 - International Conference on Fuzzy Computation
26
Table 3: Fuzzy Completion Rules for f
Z
SI
f
Z
SI Fuzzy Completion Rules
¬
-rule
Condition: {x : (¬A) n} L(x) and {x : A 1 n} / L(x)
Action: L(x) L(x) {{x : A 1 n}} and C (x) C (x) {x
A
(1 n)}
¬
-rule
Condition: {x : (¬A) n} L(x) and {x : A 1 n} / L(x)
Action: L(x) L(x) {{x : A 1 n}} and C (x) C (x) {x
A
(1 n)}
u
-rule
Condition: {x : (C
1
uC
2
) n} L(x), x is not indirectly blocked, and {{x : C
1
n},{x : C
2
n}} * L(x)
Action: L(x) L(x) {{x : C
1
n},{x : C
2
n}} and C (x) C(x) {x
C
1
n,x
C
2
n}
u
-rule
Condition: {x : (C
1
uC
2
) n} L(x), x is not indirectly blocked, and {{x : C
1
n},{x : C
2
n}} L(x) =
Action: L(x) L(x) {{x : C
1
x
1
},{x : C
2
x
2
}} and C (x) C(x) {x
C
1
x
1
,x
C
2
x
2
,x
1
+ x
2
= 1 + n,x
1
y,x
2
1 y, y {0,1},x
1
[0,1],x
2
[0,1]}
t
-rule
Condition: {x : (C
1
tC
2
) n} L(x), x is not indirectly blocked, and {{x : C
1
n},{x : C
2
n}} L(x) =
Action: L(x) L(x) {{x : C
1
x
1
},{x : C
2
x
2
}} and C(x) C (x) {x
C
1
x
1
,x
C
2
x
2
,x
1
+x
2
= n, x
1
y,x
2
1 y, y {0, 1},x
1
[0, 1], x
2
[0,1]}
t
-rule
Condition: {x : (C
1
tC
2
) n} L(x), x is not indirectly blocked, and {{x : C
1
n},{x : C
2
n}} * L(x)
Action: L(x) L(x) {{x : C
1
n},{x : C
2
n}} and C (x) C(x) {x
C
1
n,x
C
2
n}
-rule
Condition: {x : (R.C) n} L(x), x is not blocked, and x has no R
n
-neighbor y
Action: create a new node y with L(x,y) = {{[x,y] : R n}}, L(y) = {{y : C n}}, C (x,y) = {x
R
n}, and C(y) = {x
C
n}
-rule
Condition: {x : (R.C) n} L(x), x is not indirectly blocked, and x has an R
n1
R
-neighbor y with {[x,y] : R op n1} L(x,y) and {y : C n} / L(y).
Action: L(y) L(y) {{y : C n}}, if {[x,y] : R op n1} conjugates with {[x,y] : R n}, then C (y) C (y) {x
C
n}, else C(y) C (y) {x
C
n,n
1
> n}
-rule
Condition: {x : (R.C) n} L(x), x is not indirectly blocked and x has an R
n1
R
-neighbor y with {y : C n} / L(y)
Action: L(y) L(y) {{y : C n}}, if {[x,y] : R op n1} conjugates with {[x,y] : R (1 n)}, then C (y) C (y) {x
C
n}, else C(y)
C (y) {x
C
n,n1 > 1 n}
-rule
Condition: {x : (R.C) n} L(x), x is not blocked, x has no R
n
-neighbor y, and {y : C n} L(y)
Action: create a new node y with L(x,y) = {{[x,y] : R (1 n)}}, L(y) = {{y : C n}}, C (x,y) = {x
R
(1 n)}, and C (y) = {x
C
n}
,+
-rule
Condition: {x : (R.C) n} L(x), Trans(R), x is not indirectly blocked, and x has an R
n1
R
-neighbor y with {y : (R.C) n} / L(y)
Action: L(y) L(y) {{y : (R.C) n}}, if {[x, y] : R op n1} conjugates with {[x, y] : R n}, then C (y) C (y) {x
R.C
n}, else C (y)
C (y) {x
R.C
n,n1 > n}
,+
-rule
Condition: {x : (R.C) n} L(x), Trans(R), x is not indirectly blocked, and x has an R
n1
R
-neighbor y with {y : (R.C) n} / L(y)
Action: L(y) L(y) {{y : (R.C) n}}, if {[x,y] : R op n1} conjugates with {[x, y] : R (1 n)}, then C (y) C (y) {x
R.C
n}, else C (y)
C (y) {x
R.C
n,n1 > 1 n}
-rule
Condition: {A C n} T , {x : A n
1
} L(x)
Action: L(x) L(x) {{x : D n
2
}} and C (x) C(x) {x
D
n
2
,n
2
= min(n,n
1
)}
-rule
Condition: {A C n} T , {x : A n
1
} L(x)
Action: L(x) L(x) {{x : D n
2
}} and C (x) C(x) {x
D
n
2
,n
2
= min(n,n
1
)}
REASONING WITH THE FUZZY DESCRIPTION LOGIC fZSI
27
The algorithm stops when a clash occurs; KB is
consistent iff the completion rules can be applied in
such a way that they yield a complete and clash-free
completion forest, and KB is inconsistent otherwise.
Example 1. Consider a fuzzy knowledge base
KB = { CP hP.CP [0.5,1], CP(P002) [0.6,1],
(hP.CP)(P002) [0,0.4]} where we abbreviate the
concept CancerPatient and the role hasFirstDe-
greeRelatives by CP and hP, respectively. The knowl-
edge base describes that the truth degree for a first-
degree relative of a cancer patient also being a cancer
patient is greater than or equal to 0.5. Person P002 is
a cancer patient with certainty greater than 0.6 and
the possibility that one of P002’s first-degree relative
is also a cancer patient is less than or equal to 0.4.
The query is that whether KB is consistent or not.
First, because of the fuzzy concept implica-
tion axiom, {∃hP.CP(P002) [0.5,1]} is added to
A. Next, we can initialize the fuzzy tableau
by creating a node P002 and label it with
L(P002) = {{P002 : CP 0.6},{P002 : hP.CP
0.5},{P002 : hP.CP 0.4}} and C(P002) = {x
CP
0.6,x
hP.CP
0.5, x
hP.CP
0.4}}. Since both {P002 :
hP.CP 0.5} and {P002 : hP.CP 0.4} are con-
tained in the fuzzy tableau, the reasoning algorithm
obviously detects a clash. Therefore, it stops the ap-
plication of any fuzzy completion rule and returns that
KB is inconsistent.
Next, let us look at an example for the
,+
-rule.
Example 2. Consider there are two as-
sertions in a fuzzy knowledge base:
(hasFriend.Student)(John) [0.75,1] and
hasFriend(John, Mary) [0.7,1] where hasFriend is a
transitive role.
Following the preprocessing steps, we have
{John : (hasFriend.Student) 0.75} L(John)
and {[John,Mary] : hasFriend 0.7}. Since
{[John, Mary] : hasFriend 0.7} conjugates with
{[John, Mary] : hasFriend 0.25}, the
,+
-rule
is applicable, thus {Mary : (hasFriend.Student)
0.75} is added to L(Mary).
We can see from Table 3 that all these fuzzy com-
pletion rules are based on the properties and the se-
mantics of f
Z
SI concepts. Notice that since we as-
sume all concepts to be in their negation normal form,
the fuzzy concept negation rule only applies to con-
cept names.
Let us take a second look at the t
-rule and the
u
-rule. The t
-rule generates several new con-
straints {x
1
+x
2
= n, x
1
y, x
2
1 y, y {0, 1},x
1
[0,1],x
2
[0,1]}. We can see that y is an integer vari-
able with value of 0 or 1. When y = 0, we have
x
1
= 0, x
2
= n, and thus {x
C
1
0,x
C
2
n}; while
y = 1, we have x
1
= n, x
2
= 0, and thus {x
C
1
n,x
C
2
0}. These two cases are actually representing the or-
branch of the concept disjunction rule in classic DL.
That is, the {0,1} integer variable y enable the sim-
ulation of or-branching. Furthermore, by the intro-
duction of the variable y, we also transform the non-
linear constraint max(x
1
,x
2
) n into a set of linear
constraints. Similar conclusions can be drawn on the
u
-rule. Now we can see that all the fuzzy comple-
tion rules in Table 3 generate only linear constraints,
therefore, the resulted constraint set for any node or
edge is a linear constraint set. Such a property makes
it possible for the reasoning algorithm to call some
external Linear Programming solver to solve the con-
straint set.
Here is another example to explain how the rea-
soning algorithm determines the consistency of a
knowledge base.
Example 3. Consider the following fuzzy knowledge
base KB = {Trans(R), C(a) [0.7,1], D(b) [0.8,1],
R(a,b) [0.6,1], R(b,c) [0.7,1], (Inv(R).C u
Inv(R).D)(c) [0,0.5]}. We want to check the
consistency of the knowledge base.
With Trans(R) and Inv(R), we have
Trans(Inv(R)). The fuzzy tableau is initialized
as shown in Figure 1.
Figure 1: The initial fuzzy tableau of example 3.
Next, since {c : (Inv(R).C uInv(R).D) 0.5}
L(c), the u
-rule is triggered, the reasoning al-
gorithm adds {c : (Inv(R).C) x
1
} and {c :
(Inv(R).D) x
2
} to L(c), adds {x
(Inv(R).C)(c)
x
1
,x
(Inv(R).D)(c)
x
2
,x
1
+ x
2
= 1 + 0.5, x
1
y,x
2
1 y,y {0,1},x
1
[0, 1],x
2
[0, 1]} to C (c).
Next, since {c : (Inv(R).C) x
1
} L(c), {c :
(Inv(R).D) x
2
} L(c), and we have [b,c] : R
0.7, the
-rule is applicable, thus the reasoning al-
gorithm adds {b : C x
1
} and {b : D x
2
} to L(b),
adds {x
C(b)
x
1
,x
D(b)
x
2
,x
1
< 0.7,x
2
< 0.7} to
C (b). Note that the constraints x
1
< 0.7 and x
2
< 0.7
are added because of conjugation.
ICFC 2010 - International Conference on Fuzzy Computation
28
Next, since {c : (Inv(R).C) x
1
} L(c), {c :
(Inv(R).D) x
2
} L(c), we have [b,c] : R
0.7 and Trans(Inv(R)), the
,+
-rule is also ap-
plicable, thus the reasoning algorithm adds {b :
(Inv(R).C) x
1
} and {b : (Inv(R).D) x
2
} to
L(b), adds {x
(Inv(R).C)(b)
x
1
,x
(Inv(R).D)(b)
x
2
} to
C (b).
Next, since {b : (Inv(R).C) x
1
} L(b), {b :
(Inv(R).D) x
2
} L(b), and we have [a,b] : R
0.6, the
-rule is also applicable, thus the reason-
ing algorithm adds {a : C x
1
} and {a : D x
2
} to
L(a), adds {x
C(b)
x
1
,x
D(b)
x
2
,x
1
< 0.6,x
2
< 0.6}
to C (a).
Now the fuzzy tableau is shown in Figure 2.
Together with the default variable constraints, the
reasoning algorithm forms the following constraint
set:
subject to
x
C(a)
0.7,x
D(b)
0.8
x
R(a,b)
0.6,x
R(b,c)
0.7
x
(Inv(R).Cu∃Inv(R).D)(c)
0.5
x
(Inv(R).C)(c)
x
1
,x
(Inv(R).D)(c)
x
2
x
1
+ x
2
= 1 + 0.5
x
1
y,x
2
1 y
x
C(b)
x
1
,x
D(b)
x
2
x
1
< 0.7,x
2
< 0.7
x
C(a)
x
1
,x
D(a)
x
2
x
1
< 0.6,x
2
< 0.6
x
(Inv(R).C)(b)
x
1
,x
(Inv(R).D)(b)
x
2
x
C(a)
,x
D(b)
,x
R(a,b)
,x
R(b,c)
[0,1]
x
(Inv(R).Cu∃Inv(R).D)(c)
[0,1]
x
(Inv(R).C)(c)
,x
(Inv(R).D)(c)
[0,1]
y {0, 1}
x
1
,x
2
,x
C(b)
,x
D(b)
[0,1]
x
(Inv(R).C)(b)
,x
(Inv(R).D)(b)
[0,1]
Using a Linear Programming solver, e.g., the
GLPK solver (GLPK, 2008), it is easy to show that
the constraint set is unsolvable. Therefore, the fuzzy
knowledge base is inconsistent.
Through this example, it is shown that the consis-
tency check of a knowledge base can be reduced to a
problem of constraints solving in linear programming.
The constraints solving can be processed either at the
end of the reasoning procedure when no further fuzzy
completion rules are applicable, or after each applica-
tion of a completion rule. The advantage of the later
case is that, in some situations, the computation effort
could be saved when the constraints solver can iden-
tify unsolvable constraints sets earlier in the reasoning
process. However, in other situations, since calling an
external solver is time consuming, frequent calls will
severely affect the overall performance. In the former
case, we only have to call the external solver once. In
Figure 2: The extended fuzzy tableau of example 3.
addition, we can apply some optimization strategies
such as trivial clash detection and individual groups
to improve the performance.
It is well known that there is always the trade-
off issue between the expressive power of a DL and
its computational complexity. The more expressive a
DL is, the higher its computational complexity. Hor-
rocks et. al presented an optimized version of the
tableau algorithm for classic SI in (Horrocks et al.,
1998), which generates completion trees whose depth
is polynomially bounded by the size of sub(A). It is
an interesting problem to investigate the applicability
of the optimization to the fuzzy case.
4 CONCLUSIONS
In this paper, we address the fuzzy instance entail-
ment problem with respect to a fuzzy knowledge base
and then present a fuzzy extension to the expressive
Description Logic SI based on Zadeh Logic and the
residual R-implication.
For real-world applications where a knowledge
base is considered as a means to store information
(both precise and imprecise) about individuals, usu-
ally more complex inferences other than consistency
checking are required. For example, users may want
to pose a query like “Given a knowledge base, what’s
the certainty of an assertion?”. Another kind of query
can be “How many individuals belong to a given con-
cept description with a confidence greater than 0.5,
and what are they?” We describe the former query as
an instance range entailment problem and the later as
an f-retrieval problem. However, due to space limi-
tations, the reasoning methods for these problems are
REASONING WITH THE FUZZY DESCRIPTION LOGIC fZSI
29
omitted in this paper.
A prototype reasoner using SWI-Prolog and
GLPK has been under implementation based on the
ALC reasoner ALCAS (Spencer, 2006). It currently
supports functionalities to check consistency, fuzzy
instance entailment and f-retrieval of a fuzzy f
Z
ALC
knowledge base. Part of our ongoing work considers
further development of the reasoner to support other
reasoning problems as well as more expressivity in
the fuzzy knowledge base.
As we pointed out in Section 2, the properties for
transitive roles and value restrictions also hold under
Product Logic. Therefore, another direction of future
work is to investigate the reasoning algorithms for ex-
pressive fuzzy Description Logics under norms from
other logics in the family of Fuzzy Logics.
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