Why Description Logics ?
If FOL is directly used without some kind of restriction, then
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 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 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 ⊓ 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 ë“œ Person ⊓ Female Man ë“œ Person ⊓ㄱFemale
Parent ë“œ Person ⊓ ∃hasChild. Ñ‚ NotParent ë“œ 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.
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