**Category (Symbol) Construction Operators Syntax Semantics Diagrams Category (Symbol) Construction Operators Syntax Semantics Diagrams ALC** Top concept ΔI **ALC** Value restriction ∀R.C {*a* ∈ CI|∀y,(*a*, *b*)∈ *R*I & *b* ∈ CI} Bottom concept ⊥ ∅ **H** Concept inclusion C1  *C*2 *C*I 1 ⊆ *C*I 2 Atomic concept Ac *Ac*I ⊆ ΔI Role inclusion R  *S* RI ⊆ SI Atomic role R *R*I ⊆ ΔI × ΔI **I** Inverse role R− {(*a*, *b*) ∈ *R*I|(*b*, *a*) ∈ *R*I} Conjunction C D *C*I 1 ∩ *C*I 2 **R**+ Trans role Trans(R) {(*a*, *c*) ∈ *R*I|∃(*a*, *b*) ∈ *R*I ∧ (*b*, *c*) ∈ *R*I} Disjunction C D *C*I 1 ∪ *C*I 2 **Q** Qualifying at least restriction R.C ≥ n {*a* ∈ CI|#({*b*|(*a*, *b*) ∈ *R*I & *b* ∈ CI}) ≥ n} Negation ¬C ΔI\CI Qualifying at most restriction R.C ≤ n {*a* ∈ CI|#({*b*|(*a*, *b*) ∈ *R*I & *b* ∈ CI}) ≤ n} Exist restriction ∃R.C {*a* ∈ CI|∃y,(*a*, *b*)∈ *R*I & *b* ∈ CI}
