=Paper=
{{Paper
|id=Vol-218/paper-7
|storemode=property
|title=Reflections on Modelling Vagueness in Description Logics
|pdfUrl=https://ceur-ws.org/Vol-218/pos_paper1.pdf
|volume=Vol-218
|dblpUrl=https://dblp.org/rec/conf/semweb/SchockaertVHCCK06
}}
==Reflections on Modelling Vagueness in Description Logics==
https://ceur-ws.org/Vol-218/pos_paper1.pdf
Reflections on Modelling Vagueness in
Description Logics
Steven Schockaert1 , Patricia Victor1 , Geert-Jan Houben2 , Chris Cornelis1 ,
Martine De Cock1 , and Etienne Kerre1
1
Department of Applied Mathematics and Computer Science, Ghent University
2
Web & Information System Engineering Laboratory, Vrije Universiteit Brussel
Abstract. Description logics are a popular, and widely used, formalism
for describing domain knowledge in a structured way. To cope with the
vagueness of real-world knowledge, several fuzzy description logics have
already been proposed. In this contribution, we argue that, rather than
modelling all vagueness in a generalized description logic, it may be more
natural to encode vague attributes in concrete domains [1], and to model
vague predicates on the level of query processing.
Although description logics (DLs) have gained much popularity during the
past decade, applications using them suffer from one important drawback: the
information they capture is supposed to be perfect, i.e., well-defined, unambigu-
ous, certain, etc. However, real-world knowledge is often far from perfect; in
order to accommodate such imperfections, several extensions to DLs have been
proposed, including e.g. non-monotonic and probabilistic reasoning mechanisms.
In particular, we focus on techniques dealing with vagueness in DLs.
Fuzzy set theory is by far the best-known and most popular formalism to
deal with vagueness in computer science. Accordingly, a wide range of fuzzy de-
scription logics have been introduced with the aim of encoding vague knowledge
in ontologies, see e.g. [3]. In them, vague concepts are represented as mappings,
called fuzzy sets, from a suitable universe U to the unit interval [0,1]. Based
on this, fuzzy DLs allow, e.g., to express fuzzy assertional axioms, like “x is an
instance of C, at least to degree α”. Although most fuzzy DLs are based on well-
established theoretical foundations, and the main reasoning tasks are typically in
the same complexity class as their crisp counterparts, it is questionable whether
their use outside a strictly application-dependent context will ever be successful.
Mainly, this is related to the context-dependent nature of membership degrees.
After all, in a purely symbolic approach, what does it mean, e.g., to call a book
cover red to degree 0.8? For example, knowing that information, is it possible
that it is vermilion?
In our view, a more practically workable solution is to equip DLs with con-
crete domains describing vagueness. A concrete domain [1] is defined as a pair
D = (∆D , ΦD ), where ∆D is a set, and ΦD a set of predicate names, where each
name P is associated with an n-ary predicate P D ⊆ (∆D )n . A typical domain
is that of reals equipped with predicates like ≤ and =, but in general concrete
domains can be arbitrary domains satisfying some weak requirements. An im-
portant consequence of this is that in order to cope with vagueness, the idea of
concrete domains does not need to be extended; it is possible to use concrete
domains whose elements are fuzzy sets. An obvious advantage of this approach
is that we can specify exactly what we mean by “red”, for instance by means
of a fuzzy region in a color space. Note how this idea differs from a related ap-
proach by Straccia [4], whose proposal is not to allow fuzzy sets as elements of a
concrete domain, but to construct fuzzy DLs with fuzzy relations as predicates.
Furthermore, many vague concepts that arise in a semantic search setting can
be treated on the level of the query processing. For example, consider a query
asking for red books published during the late 1960s, and an ontology where colors
and time instants are modelled in a suitable concrete domain. To answer such a
vague query, we can first search for instances that satisfy a more general, crisp
query, e.g., books that are red to a strictly positive degree, published between
1965 and 1970, i.e., during the late 1960s to a strictly positive degree. Next, we
can rank the retrieved instances by the degree to which they satisfy the initial
vague query. In this way, the interpretation of vague predicates, which is typ-
ically subjective, may be based on some user profile. Moreover, while in fuzzy
description logics one operator needs to be fixed to generalize logical conjunc-
tion, treating vague predicates on the level of query processing allows for more
flexibility w.r.t. the aggregation operator that is used to combine the degrees to
which each of the criteria is satisfied.
In conclusion, DLs with concrete domains offer both a realistic and expressive
way to model objects with vague attributes. In addition, vague predicates can
conveniently be modelled at the level of query processing, drawing upon a rich
body of existing work on flexible query answering in databases (see e.g. [2]), and
making the need for fuzzy description logics, at least in the context of semantic
search, questionable.
Acknowledgment Steven Schockaert and Chris Cornelis would like to thank the
Research Foundation-Flanders for funding their research. Patricia Victor would
like to thank the Institute for the Promotion of Innovation through Science and
Technology in Flanders (IWT-Vlaanderen) for funding her research.
References
1. F. Baader, P. Hanschke, A scheme for integrating concrete domains into
concept languages, Proc. of the 12th International Joint Conference on Artificial
Intelligence, (1991), 452–457
2. H. L. Larsen, G. Pasi, D. Ortiz-Arroyo, T. Andreasen, H. Christiansen (eds.),
Proc. of the 7th International Conference on Flexible Query Answering
Systems, LNAI 4027, (2006)
3. U. Straccia, Reasoning within fuzzy description logics, Journal of Artificial
Intelligence Research (4), (2001), 137–166
4. U. Straccia, Description logics with fuzzy concrete domains, Proc. of the
21st Conference on Uncertainty in Artificial Intelligence, (2005)