Table 1 Syntax and semantics of EL++ DL

Top

Thing

⊤

∆ ^I

Bottom

Nothing

⊥

∅

Atomic class

Class

C

C^I

Primitive role

Object and data property

R

R^I

Existential quantification

ObjectSomeValuesFrom

∃R. C

{x ∈ ∆^I| ∃y ∈ ∆^I : (x, y) ∈ R ⋀ y ∈ C^I}

General class inclusion

SubClassOf

C ⊑ D

C^I ⊆ D^I

Role inclusion

SubObjectPropertyOf

R ⊑ S

{x, y ∈ ∆^I| (x, y) ∈ R^I → (x, y) ∈ S^I}

Class equivalence

EquivalentTo

C ≡ D (C ⊑ D, D ⊑ C )

C^I = D^I

Conjunction

ObjectIntersectionOf

C ⊓ D

C^I ∩ D^I

Domain restriction

ObjectPropertyDomain

∃R.  ⊤  ⊑ C

{x ∈ ∆^I| (x, y) ∈ R^I} ⊆ C^I}

Range restriction

ObjectPropertyRange

⊤ ⊑  ∀ R. C

{y ∈ ∆^I| (x, y) ∈ R^I} ⊆ C^I}

Disjointedness

DisjointWith

C ⊓ D ⊑ ⊥

C^I ∩ D^I = ∅