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Nardi, & Patel-Schneider, 2003), as the logical foundation of the standard Web Ontology Languages,
support knowledge representation and reasoning by means of the concepts and roles. The logical coun-
terparts of OWL Lite and OWL DL are the DLs SHIF
(D)
and SHOIN
(D)
, respectively. The most
prominent feature of DLs is their built-in reasoning mechanism through which implicit knowledge is
discovered from explicit information stored in a DL knowledge base (KB).
In the real world, there exists a great deal of uncertainty and imprecision which is likely the rule than
an exception. Thus, the problems that emerge are how to represent these non-crisp knowledge within
ontologies and DLs. Based on Zadeh's fuzzy set theory (Zadeh, 1965), there have been substantial
amounts of work carried out in the context of fuzzy extensions of DLs (Straccia, 2001; Stoilos, Stamou,
Pan, Tzouvaras, & Horrocks, 2007), and fuzzy ontologies (Stoilos, Simou, Stamou, & Kollias, 2006)
are thus established. For a comprehensive review of fuzzy ontologies and fuzzy DLs, the readers can
refer to (Lukasiewicz & Straccia, 2008).
Fuzzy DL reasoners (Bobillo & Straccia, 2008; Stoilos et al., 2006) implement most of the standard
fuzzy inference services (Straccia, 2001), including checking of fuzzy concept satisfiability, fuzzy con-
cept subsumption, and ABox consistency. In addition, some fuzzy DL reasoners support different kinds
of simple queries over a KB
for obtaining assertional knowledge, such as
retrieval
, i.e., given a fuzzy
KB , a fuzzy concept
C
, and
n
Î (0,1] , to retrieve all instances
o
occurring in the ABox, such that
entails
C o
( ) ³ . In fact, fuzzy DL reasoners deal with these queries by
transforming them into standard inference tasks. For example, the retrieval problem
K
C o
( ) ³ , written as
K
C o
n
n
( ) ³ can
n
be reduced to the (un)satisfiability problem of the KB È{ ( ) < }
C a
n
, while the latter one is a standard
inference problem.
With the emergence of a good number of large-scale domain ontologies encoding in OWL languages,
it is of particular importance to provide users with expressive querying service. Conjunctive queries
(CQs) originated from research in relational databases, and, more recently, have also been identified
as a desirable form of querying DL knowledge bases. Conjunctive queries provide an expressive query
language with capabilities that go beyond standard instance retrieval. For example, consider a user query
“find me hotels that are very close to the conference venue (with membership degree at lest 0.9) and
offer inexpensive (with membership degree at lest 0.7) rooms”, which can be formalized in as
Hotel x
( )
≥ ∧
1
closeTo x venue
( ,
)
≥
0.9
∧
hasRoom x y
( ,
)
≥
1
∧
hasPrice y z
( ,
)
≥ ∧ ¬
1
Expensive z
( )
≥
0.7 .
Existing DL reasoners are limited to providing basic reasoning services. There is, however, no support
for queries that ask for
n
-tuples of related individuals or for the use of variables to formulate a query,
just as conjunctive queries do. The reason for this lies in the fact that a fuzzy conjunctive query is not
expressible as a part of a fuzzy DL knowledge base. Thus a fuzzy conjunctive query entailment problem
cannot be reduced into a basic reasoning problem so as to be dealt with by existing fuzzy DL reasoners.
There is also the need for sufficient expressive power of fuzzy DLs to support reasoning in a full
fuzzy extension of the OWL Web ontology language (Stoilos et al., 2006). In this study, we thus deal
with fuzzy conjunctive query entailment for an expressive fuzzy DL
f
-SHIF (
D
, the logic counterpart
of fuzzy OWL Lite language.
