Databases Reference
In-Depth Information
semantic web community in recent years. We investigate and assess the applica-
bility and performance of our probabilistic-logical approach to data integration
using these two prominent problems. In order to make the article comprehensive,
however, we first briefly cover description logics and ontologies as these logical
concepts are needed in later parts of the document.
3.1 Ontologies and Description Logics
An Ontology usually groups objects of the world that have certain properties
in common (e.g. cities or countries) into concepts. A specification of the shared
properties that characterize a set of objects is called a concept definition. Con-
cepts can be arranged into a subclass-superclass relation in order to further
discriminate objects into subgroups (e.g. capitals or European countries). Con-
cepts can be defined in two ways, by enumeration of its members or by a concept
expression. The specific logical operators that can be used to formulate concept
expressions can vary between ontology languages.
Description logics are decidable fragments of first order logic that are de-
signed to describe concepts in terms of complex logical expressions
2
The basic
modeling elements in description logics are concepts (classes of objects), roles
(binary relations between objects) and individuals (named objects). Based on
these modeling elements, description logics contain operators for specifying so-
called concept expressions that can be used to specify necessary and sucient
conditions for membership in the concept they describe. These modeling ele-
ments are provided with a formal semantics in terms of an abstract domain
interpretation mapping
mapping each instance onto an element of an abstract
domain
Δ
I
. Instances can be connected by binary relations defined as subsets
of
Δ
I
×
I
Δ
I
. Concepts are interpreted as a subset of the abstract domain
Δ
.
Intuitively, a concept is a set of instances that share certain properties. These
properties are defined in terms of concept expressions. Typical operators are
the Boolean operators as well as universal and existential quantification over
relations to instances in other concepts.
A description logic knowledge base consists of two parts. The A-Box contains
information about objects, their type and relations between them, the so-called
T-Box consists of a set of axioms about concepts (potentially defined in terms of
complex concept expressions and relations. The first type of axioms can be used
to describe instances. In particular, axioms can be used to state that an instance
belongs to a concept or that two instances are in a certain relation. It is easy to
see, that these axioms can be used to capture case descriptions as labeled graphs.
The other types of axioms describe relations between concepts and instances. It
can be stated that one concept is a subconcept of the other (all its instances are
also instances of this other concept). Further, we can define a relation to be a
subrelation or the inverse of another relation. The formal semantics of concepts
and relations as defined by the interpretation into the abstract domain
Δ
I
can
2
Details about the relation between description logics and first-order logic can be
found in [11] and [88].
