Description Logic in AI

Why Description Logics ?

If FOL is directly used without some kind of restriction, then

• The structure of the knowledge/information is lost (no variables, concepts as classes, and roles as properties),
• The expressive power of FOL is too high for having good (computational properties and efficient procedures).

Description Logics

Description logics are a family of logics concerned with knowledge representation.

A description logic is a decidable fragment of first-order logic, associated with a set of automatic reasoning procedures.

The basic constructs for a description logic are the notion of a concept and the notion of a relationship.

Complex concept and relationship expressions can be constructed from atomic concepts and relationships with suitable constructs between them.

Example :
                    HumanMother 드 Female ㄇ∃ HasChild.Person


Axioms, Disjunctions and Negations

Teaching-Assistant ⊑ ㄱUndergrad ⊔ Professor
∀x. Teaching-Assistant(x) ➝ㄱ Undergrad(x) ⋁ Professor(x)

A necessary condition in order to be a teaching assistant is to be either not undergraduates or a professor. Clearly, a graduated student being a teaching assistant is not necessarily a professor; moreover, it may be the case that some professor is not graduated.
 
Teaching-Assistant ≐ ㄱUndergrad ⊔ Professor
∀x. Teaching-Assistant(x) ↔️ Undergrad(x) ⋁ Professor(x)

When the left-hand side is an atomic concept, the 드 symbol introduces a primitive definition (giving only necessary condition) while the ≐ symbol introduces a real definition. With necessary and sufficient conditions.
In general, it is possible to have complex concept expressions at the left-hand side as well.

 Most known description logics are :

FL - The simplest and less expressive description logic.
         C,D ➝A | C ⊓ D丨∀R.C | ∃R

ALC - A more practical and expressive description logic.
          C,D ➝ A| T | ⊥ | ㄱA | C ⊓ D丨∀R.C | ∃R.T

SHOIN - Very popular description logic .
                The logic underlying OWL.

DLR idf  - Very expressive description logic,
                  Capable of representing most database constructs.

Description logic ALC (Syntax and Semantic)

Example :

Woman 드 Person ⊓ Female       Man 드 Person ⊓ㄱFemale
Parent 드 Person ⊓ ∃hasChild. т   NotParent 드 Person ㄇ ∃hasChild.⊥

Closed Propositional Logic

Conjunction is interpreted as intersection of sets of individuals.
Disjunction is interpreted as union of sets of individuals.
Negation is interpreted as complement of sets of individuals.