Table 1 \(\boldsymbol{\mathcal {EL}}^{++}\) syntax and semantics
From: Completing the is-a structure in light-weight ontologies
Name | Syntax | Semantics |
|---|---|---|
Top | ⊤ | \(\Delta ^{{\mathcal {I}}}\) |
Bottom | ⊥ | ∅ |
Nominal | {a} | \(\{a^{{\mathcal {I}}}\}\) |
Conjunction | C⊓D | \(C^{{\mathcal {I}}} \cap D^{{\mathcal {I}}}\) |
Existential | ∃r.C | \(\left \{ x \in \Delta ^{{\mathcal {I}}} ~ | \exists y \in \Delta ^{{\mathcal {I}}} :\right.\) |
restriction | \(\left.(x,y) \in r^{{\mathcal {I}}} \wedge y \in C^{{\mathcal {I}}}\right \}\) | |
GCI | \(C {\sqsubseteq } D\) | \(C^{{\mathcal {I}}} \subseteq D^{{\mathcal {I}}}\) |
RI | \(r_{1} \circ \ldots \circ r_{k} {\sqsubseteq } r\) | \(r_{1}^{{\mathcal {I}}} \circ \ldots \circ r_{k}^{{\mathcal {I}}} \subseteq r^{{\mathcal {I}}}\) |