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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 ë“œ Female ã„‡∃ HasChild.Person

Axioms, Disjunctions and Negations

∀x. Teaching-Assistant(x) ➝ã„± 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.

∀x. Teaching-Assistant(x) ↔️ Undergrad(x) ⋁ Professor(x)

When the left-hand side is an atomic concept, the ë“œ 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 ⊓ Dä¸¨∀R.C | ∃R

ALC - A more practical and expressive description logic.
C,D ➝ A| T | ⊥ | ã„±A | C ⊓ Dä¸¨∀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 ë“œ Person ⊓ Female       Man ë“œ Person ⊓ã„±Female
Parent ë“œ Person ⊓ ∃hasChild. Ñ‚   NotParent ë“œ Person ã„‡ ∃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.