The meaning of a
terminology, and of the common inference
problems, is given in terms of a Tarski style model theoretic
semantics using
*interpretations* [2,6,4].
An interpretation
consists of a
set
, called the *abstract domain* of
, a set
, called the *concrete domain* of
, and a
*valuation function*
. The abstract and concrete
domains must be disjoint, i.e.,
.

The valuation function maps every concept to a subset of and every role to a subset of ) such that, for all concepts , , roles , , concrete predicate expressions and non-negative integers , the equations in Figure 5 are satisfied (where denotes the cardinality of a set ). Concrete predicate expressions have the obvious interpretation as shown in Figure 6. The ordering on strings is the standard lexicographic one.

The concrete domain is treated differently from the abstract domain
because it is considered to be already sufficiently structured by the
predicates **min**, **max** etc. Therefore, it is not appropriate
to form new classes of concrete objects (values) using the concept
language [1].

In order to avoid considering roles such as (i.e., the
inverse of an inverse) we will define a function
such
that
is and
is . A
role is *directly subsumed* by a role w.r.t. a
terminology
iff either
or
. A
role is *subsumed* by a role w.r.t.
(written
) iff is directly subsumed by a or
there is a role such that is directly subsumed by a and
. A role is *equivalent* to a
role w.r.t.
(written
) iff
and
.
A role is transitive in
iff
for some role such that
or
(this defines
, the set of transitive role names). A role
is a *simple* role in
iff there is no role such that
is transitive in
and
.

An interpretation
*satisfies* a
terminology
iff
for every axiom
in
,
, for
every axiom
in
,
and for
every transitive role in
,
. Such an
interpretation is called a *model* of
(written
).

A concept is satisfiable with respect to a terminology (written ) iff there is a model of with . A concept is subsumed by a concept w.r.t. (written ) holds for each model of .

An OIL ontology
is called *consistent* iff
. A class
in an ontology
is
called *consistent* iff
. A class
is a *subclass* of a class
in an
ontology
iff
.