Created on March 2, 2013, 9:02 p.m. by Hevok & updated by Hevok on May 2, 2013, 5:32 p.m.

What is Description Logics all about? How does one make calculation on top of this Logics?

For the Definition of Logic one need an interpretation of Logic. Logic must have some interpretation to make Sense of the Logic.

One has always to consider a Model Theoretical Semantics that means one tries to construct a Model that interprets the Semantics.

Here the Meanings of a description logic is given by an interpretation. This interpretation consists out of a Discourse and an Interpretation function. The Interpretation Function itself interprets all atomic Concepts and all atomic Roles or Properties will be when interpreted as a Subset of the Cartesian Product of the Domain of Discourse with itself.

Each atomic concept will be mapped or will be subset of the domain of the Discourse and also the relation of the Role or the Property.

So one tries to map the description logics to a Set Theoretical model (Set Theory). In Set Theory one has a basic set which is the domain of discourse and one maps the atoms, Properties and Roles to Subsets of this domain of Discourse.

For example the Top Concept that represents all Individuals that are in the Domain of Discourse, i.e. the most upper class of which everything is Subclass of it. In a Set Theoretical way the Interpretation of the bottom concept must be the empty set. Then one can interpret all these operators, for example the Complement or Negation can be interpreted in a way that the complement of A, the interpretation of not A is the domain of Discourse without A.

Or the Interpretation of C and D is nothing else but the Intersection of Interpretation of C and the Interpretation of D.

For the universal quantification of Role `R`

with the Range `C`

means one considers all Individuals of the Domain of a from the Discourse of where all individuals of `b`

that are connected with the Individuals of `a`

via the interpretation of the Role or Property `R`

. For them it holds `b`

must be member of Class `C`

.

On the other hand in the existential Quantification of `R`

the Range is arbitrary, so one considers all Individuals of `a`

from the universe of Discourse where there exists `a`

`b`

for all pairs `a`

and `b`

that are represented within Relation, the atomic Property of R.

**Semantics**is determined via**Interpretation**(ΔI, I)- ΔI ... Domain of Discourse ΔI ≠ ∅
- Interpretation Function: I :A -> AI ⊑ ΔI, A ... Atomic Concept I :R -> RI ⊑ ΔI x ΔI, R ... Atomic Role/Property

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