=Paper=
{{Paper
|id=Vol-423/paper-2
|storemode=property
|title=DeLorean: A Reasoner for Fuzzy OWL 1.1
|pdfUrl=https://ceur-ws.org/Vol-423/paper2.pdf
|volume=Vol-423
|dblpUrl=https://dblp.org/rec/conf/semweb/BobilloDG08b
}}
==DeLorean: A Reasoner for Fuzzy OWL 1.1==
https://ceur-ws.org/Vol-423/paper2.pdf
DeLorean: A Reasoner for Fuzzy OWL 1.1
Fernando Bobillo, Miguel Delgado, and Juan Gómez-Romero
Department of Computer Science and Artificial Intelligence, University of Granada
C. Periodista Daniel Saucedo Aranda, 18071 Granada, Spain
Phone: +34 958243194; Fax: +34 958243317
Email: fbobillo@decsai.ugr.es, mdelgado@ugr.es, jgomez@decsai.ugr.es
Abstract. Classical ontologies are not suitable to represent imprecise
or vague pieces of information, which has led to fuzzy extensions of De-
scription Logics. In order to support an early acceptance of the OWL
1.1 ontology language, we present DeLorean, the first reasoner that
supports a fuzzy extension of the Description Logic SROIQ, closely
equivalent to it. It implements some interesting optimization techniques,
whose usefulness is shown in a preliminary empirical evaluation.
1 Introduction
Ontologies are a core element in the layered architecture of the Semantic Web.
The current standard language for ontology representation is the Web Ontology
Language (OWL). However, since its first development, several limitations on the
expressiveness of OWL have been identified, and consequently several extensions
to the language have been proposed. Among them, the most significant is OWL
1.1 [1] which is its most likely immediate successor. Description Logics (DLs
for short) [2] are a family of logics for representing structured knowledge. They
have proved to be very useful as ontology languages, and the DL SROIQ(D)
is actually closely equivalent to OWL 1.1.
It has been widely pointed out that classical ontologies are not appropriate
to deal with imprecise and vague knowledge, which is inherent to several real-
world domains. Since fuzzy logic is a suitable formalism to handle these types
of knowledge, several fuzzy extensions of DLs have been proposed [3].
The broad acceptance of the forthcoming OWL 1.1 ontology language will
largely depend on the availability of editors, reasoners, and other numerous tools
that support the use of OWL 1.1 from a high-level/application perspective [4].
With this idea in mind, this work reports the implementation of DeLorean,
the first reasoner that supports the fuzzy DL SROIQ. We also present a new
optimization (handling superfluous elements before applying crisp reasoning)
and a preliminary evaluation of the optimizations in the reduction.
This paper is organized as follows. Section 2 describes the fuzzy DL SROIQ,
which is equivalent to the fuzzy language supported by DeLorean. Then,
Section 3 describes the reasoning algorithm, based on a reduction into crisp
SROIQ. Section 4 presents our implementation, some of the implemented opti-
mizations (including handling of superfluous elements), and a preliminary eval-
uation. Finally, Section 6 sets out some conclusions and ideas for future work.
2 A Quick View to Fuzzy SROIQ
In this section we recall the definition of fuzzy SROIQ [6], which extends
SROIQ to the fuzzy case by letting concepts denote fuzzy sets of individuals
and roles denote fuzzy binary relations. Axioms are also extended to the fuzzy
case and some of them hold to a degree. We will assume a set of degrees of
truth which are rational numbers of the form α ∈ (0, 1], β ∈ [0, 1) and γ ∈ [0, 1].
Moreover, we will assume a set of inequalities $% ∈ {≥, <, ≤, >}, ! ∈ {≥, <},
" ∈ {≤, >}. For every operator $%, we define: (i) its symmetric operator $%− ,
defined as ≥− =≤, >− =<, ≤− =≥, <− =>; (ii) its negation operator ¬ $%, defined
as ¬ ≥=<, ¬ >=≤, ¬ ≤=>, ¬ <=≥.
Syntax. In fuzzy SROIQ we have three alphabets of symbols, for concepts
(C), roles (R), and individuals (I). The set of roles is defined by RA ∪ U ∪
{R− |R ∈ RA }, where RA ∈ R, U is the universal role and R− is the inverse
of R. Assume A ∈ C, R, S ∈ R (where S is a simple role [7]), oi ∈ I for
1 ≤ i ≤ m, m ≥ 1, n ≥ 0. Fuzzy concepts are defined inductively as follows:
C, D → & | ⊥ | A | C ( D | C ) D | ¬C | ∀R.C | ∃R.C | {α1 /o1 , . . . , αm /om } | (≥
m S.C) | (≤ n S.C) | ∃S.Self . Let a, b ∈ I. The axioms in a fuzzy Knowledge
Base K are grouped in a fuzzy ABox A, a fuzzy TBox T , and a fuzzy RBox R1
as follows:
ABox
Concept assertion !a : C ≥ α#, !a : C > β#, !a : C ≤ β#, !a : C < α#
Role assertion !(a, b) : R ≥ α#, !(a, b) : R > β#, !(a, b) : R ≤ β#, !(a, b) : R < α#
Inequality assertion !a %= b#
Equality assertion !a = b#
TBox
Fuzzy GCI !C & D ≥ α#, !C & D > β#
Concept equivalence C ≡ D, equivalent to {!C & D ≥ 1#, !D & C ≥ 1#}
RBox
Fuzzy RIA !R1 R2 . . . Rn & R ≥ α#, !R1 R2 . . . Rn & R > β#
Transitive role axiom trans(R)
Disjoint role axiom dis(S1 , S2 )
Reflexive role axiom ref (R)
Irreflexive role axiom irr(S)
Symmetric role axiom sym(R)
Asymmetric role axiom asy(S)
Semantics. A fuzzy interpretation I is a pair (∆I , ·I ), where ∆I is a non empty
set (the interpretation domain) and ·I a fuzzy interpretation function mapping:
(i) every individual a onto an element aI of ∆I ; (ii) every concept C onto a
function C I : ∆I → [0, 1]; (iii) every role R onto a function RI : ∆I × ∆I →
[0, 1]. C I (resp. RI ) denotes the membership function of the fuzzy concept C
1
The syntax of role axioms is restricted to guarantee the decidability of the logic [6].
(resp. fuzzy role R) w.r.t. I. C I (x) (resp. RI (x, y)) gives us the degree of being
the individual x an element of the fuzzy concept C (resp. the degree of being
(x, y) an element of the fuzzy role R) under the fuzzy interpretation I. Given
a t-norm ⊗, a t-conorm ⊕, a negation function / and an implication function
⇒ [8], the interpretation is extended to complex concepts and roles as:
&I (x) = 1
⊥I (x) = 0
(C ( D)I (x) = C I (x) ⊗ DI (x)
(C ) D)I (x) = C I (x) ⊕ DI (x)
(¬C)I (x) = /C I (x)
(∀R.C)I (x) = inf y∈∆I {RI (x, y) ⇒ C I (y)}
(∃R.C)I (x) = supy∈∆I {RI (x, y) ⊗ C I (y)}
{α1 /o1 , . . . , αm /om }I (x) = supi | x=oIi αi
!
(≥ m S.C)I (x) = supy1 ,...,ym ∈∆I [(⊗ni=1 {S I (x, yi ) ⊗ C I (yi )}) (⊗j 0 then S I (y, x) = 0,
– a fuzzy KB K = 3A, T , R4 iff it satisfies each element in A, T and R.
Irreflexive, transitive and symmetric role axioms are syntactic sugar for every
R-implication (and consequently it can be assumed that they do not appear in
fuzzy KBs) due to the following equivalences: irr(S) ≡ 3& 5 ¬∃S.Self ≥ 14,
trans(R) ≡ 3RR 5 R ≥ 14 and sym(R) ≡ 3R 5 R− ≥ 14.
In the rest of the paper we will only consider fuzzy KB satisfiability, since
(as in the crisp case) most inference problems can be reduced to it [9].
3 A Crisp Representation for Fuzzy SROIQ
In this section we show how to reduce a fuzzy Z SROIQ fuzzy KB into a
crisp KB (see [5, 6] for details). The procedure preserves reasoning, in such a
way that existing SROIQ reasoners could be applied to the resulting KB. The
basic idea is to create some new crisp concepts and roles, representing the α-cuts
of the fuzzy concepts and roles, and to rely on them. Next, some new axioms
are added to preserve their semantics, and finally every axiom in the ABox, the
TBox and the RBox is represented using these new crisp elements.
Adding new elements. Let AK and RK be the set of atomic concepts and
roles occurring in a fuzzy KB K = 3A, T , R4. The set of the degrees which must
be considered for any reasoning task is defined as N K = γ, 1 − γ | 3τ $% γ4 ∈ K.
Now, for each α, β ∈ N K with α ∈ (0, 1] and β ∈ [0, 1), for each A ∈ AK
and for each RA ∈ RK , two new atomic concepts A≥α , A>β and two new atomic
roles R≥α , R>β are introduced. A≥α represents the crisp set of individuals which
are instance of A with degree higher or equal than α i.e the α-cut of A.
The semantics of these newly introduced atomic concepts and roles is pre-
served by some terminological and role axioms. For each 1 ≤ i ≤ |N K | − 1, 2 ≤
j ≤ |N K |−1 and for each A ∈ AK , T (N K ) is the smallest terminology containing
these two axioms: A≥γi+1 5 A>γi , A>γj 5 A≥γj . Similarly, for each RA ∈ RK ,
R(N K ) contains these axioms: R≥γi+1 5 R>γi , R>γi 5 R≥γi .
Example 1. Consider the fuzzy KB K = {τ }, where τ = 3StGenevieveTexasWhite :
WhiteWine ≥ 0.754}. We have that N K = {0, 0.25, 0.5, 0.75, 1} and T (N K ) =
{WhiteWine≥0.25 5 WhiteWine>0 , WhiteWine>0.25 5 WhiteWine≥0.25 , WhiteWi-
ne≥0.5 5 WhiteWine>0.25 , WhiteWine>0.5 5 WhiteWine≥0.5 , WhiteWine≥0.5 5
WhiteWine>0.25 , WhiteWine>0.5 5 WhiteWine≥0.5 , WhiteWine≥0.75 5 WhiteWi-
ne>0.5 , WhiteWine>0.75 5 WhiteWine≥0.75 , WhiteWine≥1 5 WhiteWine>0.75 }. ()
Mapping fuzzy concepts, roles and axioms. Concept and role expressions
are reduced using mapping ρ, as shown in the first part of Table 1. Given a fuzzy
concept C, ρ(C, ≥ α) is a crisp set containing all the elements which belong to C
with a degree greater or equal than α (the other cases are similar). For instance,
the 1-cut of the fuzzy concept ∀madeFromFruit.(NonSweetFruit ) SweetFruit) is
ρ(∀madeFromFruit.(NonSweetFruit ) SweetFruit), ≥ 1) = ∀madeFromFruit>0 .Non-
SweetFruit≥1 ) SweetFruit≥1 .
Finally, we map the axioms in the ABox, TBox and RBox. Axioms are re-
duced as shown in the second part of Table 1, where σ maps fuzzy axioms into
crisp assertions, and κ maps fuzzy TBox (resp. RBox) axioms into crisp TBox
(resp. RBox) axioms. Recall that we are assuming that irreflexive, transitive
and symmetric role axioms do not appear in the RBox. For example, assuming
N K = {0, 0.25, 0.5, 0.75, 1}, the reduction of the fuzzy GCI 3Port 5 RedWine ≥ 14
is κ(3Port 5 RedWine ≥ 14) = {Port>0 5 RedWine>0 , Port≥0.25 5 RedWine≥0.25 ,
Port>0.25 5 RedWine>0.25 , Port≥0.5 5 RedWine≥0.5 , Port>0.5 5 RedWine>0.5 ,
Port≥0.75 5 RedWine≥0.75 , Port>0.75 5 RedWine>0.75 , Port≥1 5 RedWine≥1 }.
Table 1. Mapping of concept and role expressions, and reduction of the axioms. The
semantics of &G uses Gödel implication, that of &KD uses Kleene-Dienes implication.
Fuzzy concepts
ρ((, !γ) (
ρ((, "γ) ⊥
ρ(⊥, !γ) ⊥
ρ(⊥, "γ) (
ρ(A, !γ) A!γ
ρ(A, "γ) ¬A¬"γ
ρ(¬C, %& γ) ρ(C, %&− 1 − γ)
ρ(C + D, !γ) ρ(C, !γ) + ρ(D, !γ)
ρ(C + D, "γ) ρ(C, "γ) , ρ(D, "γ)
ρ(C , D, !γ) ρ(C, !γ) , ρ(D, !γ)
ρ(C , D, "γ) ρ(C, "γ) + ρ(D, "γ)
ρ(∃R.C, !γ) ∃ρ(R, !γ).ρ(C, !γ)
ρ(∃R.C, "γ) ∀ρ(R, ¬" γ).ρ(C, "γ)
ρ(∀R.C, {≥, >}γ) ∀ρ(R, {>, ≥}1 − γ).ρ(C, {≥, >}γ)
ρ(∀R.C, "γ) ∃ρ(R, "− 1 − γ).ρ(C, "γ)
ρ({α1 /o1 , . . . , αm /om }, %& γ) {oi | αi %& γ, 1 ≤ i ≤ m}
ρ(≥ m S.C, !γ) ≥ m ρ(S, !γ).ρ(C, !γ)
ρ(≥ m S.C, "γ) ≤ m−1 ρ(S, ¬ " γ).ρ(C, ¬ " γ)
ρ(≤ n S.C, {≥, >} γ) ≤ n ρ(S, {>, ≥} 1 − γ).ρ(C, {>, ≥} 1 − γ)
ρ(≤ n S.C, "γ) ≥ n+1 ρ(S, "− 1 − γ).ρ(C, "− 1 − γ)
ρ(∃S.Self, !γ) ∃ρ(S, !γ).Self
ρ(∃S.Self, "γ) ¬∃ρ(S, ¬ " γ).Self
Fuzzy roles
ρ(RA , !γ) RA!γ
ρ(RA , "γ) ¬RA¬"γ
ρ(R− , %& γ) ρ(R, %& γ)−
ρ(U, !γ) U
ρ(U, "γ) ¬U
Axioms
σ(!a : C %& γ#) {a : ρ(C, %& γ)}
σ(!(a, b) : R %& γ#) {(a, b) : ρ(R, %& γ)}
σ(!a %= b#) {a %= b}
σ(!a = b#) {a = b}
!
κ(C &G D ≥ α) {ρ(C, ≥ γ) & ρ(D, ≥ γ)}
!γ∈N f K \{0} | γ≤α
γ∈N f K | γ<α {ρ(C, > γ) & ρ(D, > γ)}
κ(C &G D > β) κ(C & D ≥ β) ∪ {ρ(C, > β) & ρ(D, > β)}
κ(C &KD D ≥ α) {ρ(C, > 1 − α) & ρ(D, ≥ α) }
κ(C &KD D > β) {ρ(C, ≥ 1 − β) & ρ(D, > β) }
!
κ(!R1 . . . Rn &G R ≥ α#) {ρ(R1 , ≥ γ) . . . ρ(Rn , ≥ γ) & ρ(R, ≥ γ)}
!γ∈N f K \{0} | γ≤α
γ∈N f K | γ<α {ρ(R 1 , > γ) . . . ρ(Rn , > γ) & ρ(R, > γ)}
κ(!R1 . . . Rn &G R > β#) κ(!R1 . . . Rn & R ≥ β#) ∪
{ρ(R1 , > β) . . . ρ(Rn , > β) & ρ(R, > β)}
κ(!R1 . . . Rn &KD R ≥ α#) {ρ(R1 , > 1 − α) . . . ρ(Rn , > 1 − α) & ρ(R, ≥ α)}
κ(!R1 . . . Rn &KD R > β#) {ρ(R1 , ≥ 1 − β) . . . ρ(Rn , ≥ 1 − β) & ρ(R, > β)}
κ(dis(S1 , S2 )) {dis(ρ(S1 , > 0), ρ(S2 , > 0))}
κ(ref (R)) {ref (ρ(R, ≥ 1))}
κ(asy(S)) {asy(ρ(S, > 0)}
Properties. Summing up, a fuzzy KB K = 3A, T , R4 is reduced into a KB
crisp(K) = 3σ(A), T (N K ) ∪ κ(K, T ), R(N K ) ∪ κ(K, R)4. The following theorem
shows that the reduction preserves reasoning:
Theorem 1. A Z SROIQ fuzzy KB K is satisfiable iff crisp(K)) is [6].
The resulting KB is quadratic because it depends on the number of relevant
degrees |N K |, or linear if we assume a fixed set. An interesting property is that
the reduction of an ontology can be reused when adding a new axiom. If the new
axioms does not introduce new atomic concepts, atomic roles nor a new degree
of truth, we just need to add the reduction of the axiom.
4 DeLorean Reasoner
This section describes the prototype implementation of our reasoner, which
is called DeLorean (DEscription LOgic REasoner with vAgueNess).
Initially, we developed a first version based on Jena API2 [6]. This version
was developed in Java, using the parser generator JavaCC3 , and DIG 1.1 in-
terface [10] to communicate with crisp DL reasoners. An interesting property is
the possibility of using any crisp reasoner thanks to the DIG interface. However,
DIG interface does not yet support full SROIQ, so the logic supported by De-
Lorean was restricted to Z SHOIN (OWL DL). From a historical point of
view, this version was the first reasoner that supported a fuzzy extension of the
OWL DL language. It implemented the reduction described in [11], and applied
the optimization in the number of new elements and axioms described below.
With the aim of augmenting the expressivity of the logic, in the current
version we have changed the subjacent API to OWL API for OWL 24 [4]. Now,
DeLorean supports both Z SROIQ(D) and G SROIQ(D), which correspond
to fuzzy versions of OWL 1.1 under Zadeh and Gödel semantics, respectively.
Since DIG interface does not currently allow the full expressivity of OWL
1.1, our solution was to integrate directly DeLorean with a concrete crisp
ontology reasoner: Pellet [12], which can be directly used from the current
version of the OWL API. This way, the user is free to choose to use either a
generic crisp reasoner (restricting the expressivity to SHOIQ) or Pellet with
no expressivity limitations. DeLorean is the first reasoner that supports a fuzzy
extension of OWL 1.1.
Figure 1 illustrates the architecture of the system:
– The Parser reads an input file with a fuzzy ontology and translates it into
an internal representation. The point here is that we can use any language
to encode the fuzzy ontology, as long as the Parser can understand the
representation and the reduction is properly implemented. Consequently we
will not get into details of our particular choice.
2
http://jena.sourceforge.net/
3
https://javacc.dev.java.net
4
http://owlapi.sourceforge.net
Fig. 1. Architecture of DeLorean reasoner.
In order to make the representation of fuzzy KBs easier, DeLorean also al-
lows the possibility of importing OWL 1.1 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 (Zadeh or Gödel family) for some complex concept.
– The Reduction module implements the reduction procedures described in the
previous section, building an OWL API model with an equivalent crisp on-
tology which can be exported to an OWL file. The implementation also takes
into account all the optimizations already discussed along this document.
– The Inference module tests this ontology for consistency, using either Pel-
let or any crisp reasoner through the DIG interface. Crisp reasoning does
not take into account superfluous elements as we explain below.
– Inputs (the path of the fuzzy ontology) and outputs (the result of the rea-
soning and the elapsed time) are managed by an User interface.
The reasoner implements the following optimizations:
Optimizing the number of new elements and axioms. Previous works
use two more atomic concepts A≤β , A<α and some additional axioms A<γk 5
A≤γk , A≤γi 5 A<γi+1 , A≥γk (A<γk 5 ⊥, A>γi (A≤γi 5 ⊥, & 5 A≥γk )A<γk , & 5
A>γi ) A≤γi , 2 ≤ k ≤ |N K |. In [6] it is shown that they are unnecessary.
Optimizing GCI reductions. In some particular cases, the reduction of fuzzy
GCIs can be optimized [6]. For example, in range role axioms of the form 3& 5
∀R.C ≥ 14, domain role axioms of the form 3& 5 ∀R− .C ≥ 14 and functional
role axioms of the form 3& 5≤ 1 R.& ≥ 14 we can use that κ(3& 5 D $% γ4) =
& 5 ρ(D, $% γ). Also, in disjoint concept axioms of the form 3C ( D 5 ⊥ ≥ 14,
we can use that κ(C 5 ⊥ $% γ) = ρ(C, > 0) 5 ⊥. Furthermore, if the resulting
TBox contains A 5 B, A 5 C and B 5 C, then A 5 C is unnecessary since it
can be deduced from the other two axioms.
Allowing crisp concepts and roles. Suppose that A is a fuzzy concept. Then,
we need N K − 1 concepts of the form A≥α and another N K − 1 concepts of the
form A>β to represent it, as well as 2 · (|N K | − 1) − 1 axioms to preserve their
semantics. Fortunately, in real applications not all concepts and roles will be
fuzzy. If A is declared to be crisp, we just need one concept to represent it and
no new axioms. The case for fuzzy roles is exactly the same. Of course, this
optimization requires some manual intervention.
Reasoning ignoring superfluous elements. Our reduction is designed to
promote reusing. For instance, consider the fuzzy KB K in Example 1. The reduc-
tion of K contains σ(τ ) = StGenevieveTexasWhite : WhiteWine≥0.75 , but also the
axioms in T (N K ). It can be seen that the concepts WhiteWine>0 , WhiteWine≥0.25 ,
WhiteWine>0.25 , WhiteWine≥0.5 , WhiteWine>0.5 , WhiteWine>0.75 , WhiteWine≥1 are
superfluous in the sense that cannot cause a contradiction. Hence, for a satisfia-
bility test of crisp(K), we can avoid the axioms in T (N K ) where they appear.
But please note that if additional axioms are added to K, crisp(K) will
be different and previous superfluous concept and roles may not be superfluous
any more. For example, if we want to check if K ∪ 3StGenevieveTexasWhite :
WhiteWine ≥ 0.54 is satisfiable, then the concept WhiteWine≥0.5 is no longer
superfluous. In this case, it is enough to consider T % (N K ) = {WhiteWine≥0.75 5
WhiteWine≥0.5 }. The case of atomic roles is similar to that of atomic concepts.
5 Use Case: A Fuzzy Wine Ontology
This section considers a concrete use case, a fuzzy extension of the well-known
Wine ontology5 , a highly expressive ontology (in SHOIN (D)). Some metrics of
the ontology are shown in the first column of Table 2. In an empirical evaluation
of the reductions of fuzzy DLs to crisp DLs, P. Cimiano et al. wrote that “the
Wine ontology showed to be completely intractable both with the optimized
and unoptimized reduction even using only 3 degrees” [13]. They only considered
there what we have called here “optimization of the number of new elements and
axioms”. We will show that the rest of the optimizations, specially the (natural)
assumption that there are some crisp elements, reduce significantly the number
of axioms, even if tractability of the reasoning is to be verified.
A fuzzy extension of the ontology. We have defined a fuzzy version of the
Wine ontology by adding a degree to the axioms. Given a variable set of degrees
N K , the degrees of the truth for fuzzy assertions is randomly chosen in N K . In
the case of fuzzy GCIs and RIAs, the degree is always 1 in special GCIs (namely
concept equivalences and disjointness, domain, range and functional role axioms)
or if there is a crisp element in the left side; otherwise, the degree is 0.5.
In most of the times fuzzy assertions are of the form 3τ ! β4 with β 1= 1.
Clearly, this favors the use of elements of the forms C!β and R!β , reducing the
number of superfluous concepts. Once again, we are in the worst case from the
5
http://www.w3.org/TR/2003/CR-owl-guide-20030818/wine.rdf
point of view of the size of the resulting crisp ontology. Nonetheless, in practice
we will be often able to say that an individual fully belongs to a fuzzy concept,
or that two individuals are fully related by means of a fuzzy role.
Crisp concepts and roles. A careful analysis of the fuzzy KB brings about
that most of the concepts and the roles should indeed be interpreted as crisp.
For example, most of the subclasses of the class Wine refer to a well-defined
geographical origin of the wines. For instance, Alsatian wine is a wine which
has been produced in the French region of Alsace: AlsatianWine ≡ Wine (
∃locatedAt.{alsaceRegion}. In other applications there could exist examples of
fuzzy regions, but this is not our case. Another important number of sub-
classes of Wine refer to the type of grape used, which is also a crisp concept.
For instance, Riesling is a wine which has been produced from Riesling grapes:
Riesling ≡ Wine ( ∃madeFromGrape.{RieslingGrape}( ≥ 1 madeFromGrape.&.
Clearly, there are other concepts with no sharp boundaries (for instance,
those derived from the vague terms “dry”, “sweet”, “white” or“heavy”). The re-
sult of our study has identified 50 fuzzy concepts in the Wine ontology, namely:
WineColor, RedWine, RoseWine, WhiteWine, RedBordeaux, RedBurgundy, RedTa-
bleWine, WhiteBordeaux, WhiteBurgundy, WhiteLoire, WhiteTableWine, WineSugar,
SweetWine, SweetRiesling, WhiteNonSweetWine, DryWine, DryRedWine, DryRies-
ling, DryWhiteWine, WineBody, FullBodiedWine, WineFlavor, WineTaste, LateHar-
vest, EarlyHarvest, NonSpicyRedMeat, NonSpicyRedMeatCourse, SpicyRedMeat,
PastaWithSpicyRedSauce, PastaWithSpicyRedSauceCourse, PastaWithNonSpicyRed-
Sauce, PastaWithNonSpicyRedSauceCourse, SpicyRedMeatCourse, SweetFruit, Sweet-
FruitCourse, SweetDessert, SweetDessertCourse, NonSweetFruit, NonSweetFruit-
Course, RedMeat, NonRedMeat, RedMeatCourse, NonRedMeatCourse, PastaWith-
HeavyCreamSauce, PastaWithLightCreamSauce, Dessert, CheeseNutsDessert, De-
ssertCourse, CheeseNutsDessertCourse, DessertWine.
Furthermore, we identified 5 fuzzy roles: hasColor, hasSugar, hasBody, hasFla-
vor, and hasWineDescriptor (which is a super-role of the other four).
Measuring the importance of the optimizations. The reduction under
Gödel semantics is still to be published [14], so we focus our experimentation in
Z SROIQ (omitting the concrete role yearValue), but allowing the use of both
Kleene-Dienes and Gödel implications in the semantics of fuzzy GCIs and RIAs.
Table 2 shows some metrics of the crisp ontologies obtained in the reduction
of the fuzzy ontology after applying different optimizations.
1. Column “Original” shows some metrics of the original ontology.
2. “None” considers the reduction obtained after applying no optimizations.
3. “(NEW)” considers the reduction obtained after optimizing the number of
new elements and axioms.
4. “(GCI)” considers the reduction obtained after optimizing GCI reductions.
5. “(C/S)” considers the reduction obtained after allowing crisp concepts and
roles and ignoring superfluous elements.
6. Finally, “All” applies all the previous optimizations.
Table 2. Metrics of the Wine ontology and its fuzzy versions using 5 degrees
Original None (NEW) (GCI) (C/S) All
Individuals 206 206 206 206 206 206
Named concepts 136 2176 486 2176 800 191
Abstract roles 16 128 128 128 51 20
Concept assertions 194 194 194 194 194 194
Role assertions 246 246 246 246 246 246
Inequality assertions 3 3 3 3 3 3
Equality assertions 0 0 0 0 0 0
New GCIs 0 4352 952 4352 1686 324
Subclass axioms 275 1288 1288 931 390 390
Concept equivalences 87 696 696 696 318 318
Disjoint concepts 19 152 152 19 152 19
Domain role axioms 13 104 104 97 104 97
Range role axioms 10 80 80 10 80 10
Functional role axioms 6 48 48 6 48 6
New RIAs 0 136 119 136 34 34
Sub-role axioms 5 40 40 40 33 33
Role equivalences 0 0 0 0 0 0
Inverse role axioms 2 16 16 16 2 2
Transitive role axioms 1 8 8 8 1 1
We have put together the optimizations of crisp and superfluous elements
because in this ontology handling superfluous concepts is not always useful, due
to the existence of a lot of concept definitions, as we will see in the next example.
Example 2. Consider the fuzzy concept NonRedMeat. Firstly, this concept ap-
pears as part of a fuzzy assertion stating that pork is a non read meat: σ(3Pork :
NonRedMeat ! α1 4) = Pork : NonRedMeat!α1 . Secondly, non read meat is de-
clared to be disjoint from read meat: κ(3RedMeat ( NonRedMeat 5 ⊥ ≥ 14) =
RedMeat>0 ( NonRedMeat>0 5 ⊥. Thirdly, non read meat is a kind of meat:
κ(3NonRedMeat 5 Meat ≥ α2 4) = NonRedMeat>0 5 Meat. If these were the
only occurrences of NonRedMeat, then the reduction would create only two non-
superfluous crisp concepts, namely NonRedMeat>0 and NonRedMeat!α1 , and in
order to preserve the semantics of them we would need to add just one axiom
during the reduction: NonRedMeat!α1 5 NonRedMeat>0 .
However, this is not true because NonRedMeat appears in the definition of the
fuzzy concept NonRedMeatCourse. In fact, κ(NonRedMeatCourse ≡ MealCourse
( ∀hasFood.NonRedMeat) introduces non-superfluous crisp concepts for the rest
of the degrees in N K . Consequently, for each 1 ≤ i ≤ |N K | − 1, 2 ≤ j ≤
|N K |−1, the reduction adds to T (N K ) the following axioms: NonRedMeat≥γi+1 5
NonRedMeat>γi ; NonRedMeat>γj 5 NonRedMeat≥γj . (
)
Note that the size of the ABox is always the same, because every axiom in
the fuzzy ABox generates exactly one axiom in the reduced ontology.
The number of new GCIs and RIAs added to preserve the semantics of the
new elements is much smaller in the optimized versions. In particular, we reduce
from 4352 to 324 GCIs (7.44%) and from 136 to 34 RIAs (25%). This shows the
importance of reducing the number of new crisp elements and their corresponding
axioms, as well as of defining crisp concepts and roles and (to a lesser extent)
handling superfluous concepts.
Optimizing GCI reductions turns out to be very useful in reducing the num-
ber of disjoint concepts, domain, range and functional role axioms: 152 to 19
(12.5 %), 104 to 97 (93.27 %), 80 to 10 (12.5 %), and 48 to 6 (12.5 %), respec-
tively. In the case of domain role axioms the reduction is not very high because
we need an inverse role to be defined in order to apply the reduction, and this
happens only in one of the axioms.
Every fuzzy GCI or RIA generates several axioms in the reduced ontology.
Combining the optimization of GCI reductions with the definition of crisp con-
cepts and roles reduces the number of new axioms, from 1288 to 390 subclass
axioms (30.28 %), from 696 to 318 concept equivalences (45.69 %) and from 40
to 33 sub-role axioms (82.5 %).
Finally, the number of inverse and transitive role axioms is reduced in the
optimized version because fuzzy roles interpreted as crisp introduce one inverse
or transitive axiom instead of several ones. This allows a reduction from 16 to 2
axioms, and from 8 to 1, respectively, which corresponds to the 12.5 %.
Table 3 shows the influence of the number of degrees on the size of the
resulting crisp ontology, as well as on the reduction time (which is shown in
seconds), when all the described optimizations are used. The reduction time is
small enough to allow to recompute the reduction of an ontology when necessary,
thus allowing superfluous concepts and roles in the reduction to be avoided.
Table 3. Influence of the number of degrees in the reduction.
Crisp 3 5 7 9 11 21
Number of axioms 811 1166 1674 2182 2690 3198 5738
Reduction time - 0.343 0.453 0.64 0.782 0.859 1.75
6 Conclusions and Future Work
This paper has presented DeLorean, the more expressive fuzzy DL reasoner
that we are aware of (it supports fuzzy OWL 1.1), and the optimizations that
it implements. Among them, the current version enables the definition of crisp
concepts and roles, as well as handling superfluous concepts and roles before
applying crisp reasoning. A preliminary evaluation shows that these optimiza-
tions help to reduce significantly the size of the resulting ontology. In future
work we plan to develop a more detailed benchmark by relying on the hyper-
tableau reasoner HermiT, which seems to outperform other DL reasoners [15],
and, eventually, to compare it against other fuzzy DL reasoners.
Acknowledgements
This research has been partially supported by the project TIN2006-15041-C04-01 (Mi-
nisterio de Educación y Ciencia). Fernando Bobillo holds a FPU scholarship from Minis-
terio de Educación y Ciencia. Juan Gómez-Romero holds a scholarship from Consejerı́a
de Innovación, Ciencia y Empresa (Junta de Andalucı́a).
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