Some useful information (from http://en.wikipedia.org/wiki/Description_logic)

Syntax of description logics consists of

• A set of unary predicate symbols that are used to denote concept names;
• A set of binary relations that are used to denote role names;
• A recursive definition for defining concept terms from concept names and role names using constructors.

In description logics, concept names are regarded as atomic concepts, role names are regarded as atomic roles. In general, a concept denotes the set of individuals that belongs to it, and a role denotes a relationship between concepts.

The semantics of description logics is defined by interpreting concepts as sets of individuals and roles as sets of pairs of individuals. Those individuals are typically assumed from a given domain. The semantics of non atomic concepts and roles is then defined in terms of atomic concepts and roles. This is done by using a recursive definition similar to the syntax.

For example, given a set as the domain, an interpretation of AL-concepts is defined first over atomic concepts and roles as follows:

• An atomic concept is interpreted as a set of individuals that is a subset of the domain.
• An atomic role is interpreted as a set of pairs of individuals from the domain, i.e., a binary relation over the domain. In this case, if an individual x is related to y via a role R, then y is called an R-successor of x.

Next, this interpretation is extended to non atomic concept and role according to the constructors. This is done in the following.

• The top concept is interpreted as the whole domain.
• The bottom concept is interpreted as the empty set.
• The interpretation of ¬C is the set of all individuals in the domain which does not belong to the interpretation of C.
• Intersection of two concepts C and D is interpreted as set-intersection, i.e., the set of all individuals in the domain that belongs to both the interpretation of C and the interpretation of D.
• The value restriction ∀R.C is interpreted as the set of all individuals in the domain whose R-successors (if any) all belong to the interpretation of C.
• The limited existential restriction ∃R.Τ is interpreted as the set of all individuals in the domain that have at least one R-successor.

In DLs, a distinction is drawn between the so-called TBox (terminological box) and the ABox (assertional box). In general, the TBox contains sentences describing concept hierarchies (i.e., relations between concepts) while the ABox contains „ground” sentences stating where in the hierarchy individuals belong (i.e., relations between individuals and concepts).

A concept in DL jargon is referred to as a class in OWL. A role in DL jargon is a property in OWL.