Foundations of Semantic Web Technologies

174 Foundations of Semantic Web Technologies

• ≤nR.C describes those things which are conne cted via R to at most n

things in C, i.e.

8

(≤nR.C)

I

= {x | #{(x, y) ∈ R

I

| y ∈ C

I

} ≤ n}.

• ≥nR.C describes those things which are connected via R to at least n

things in C, i.e. (≥nR.C)

I

= {x | #{(x, y) ∈ R

I

| y ∈ C

I

} ≥ n}.

• {a} describes the class containing only a, i.e. {a}

I

= {a

I

}.

• ∃S.Self describes those things which are connected to themselves via S,

i.e. (∃S.Self)

I

= {x | (x, x) ∈ S

I

}.

• For R ∈ R, we set (R

−

)

I

= {(b, a) | (a, b) ∈ R

I

}.

• For the universal role U, we set U

I

= ∆ × ∆.

Given a SROIQ knowledge base, an interpretation consists of a domain ∆

and an