Slot constraints

A slot-constraint consists of a slot name $\textsf{\textsl{SN}}$ followed by one or more constraints that apply to the slot, written $\ensuremath{\textsf{\textsl{SN}}}[a_1,\ldots,a_n]$. Each constraint can be either:

• A value constraint with either a list of one or more class-expressions or a list of one or more concrete type expressions, written $\exists C_1,\ldots,C_n$.
• A value-type constraint with either a list of one or more class-expressions or a list of one or more concrete type expressions, written $\forall C_1,\ldots,C_n$.
• A has-filler constraint with either a list of one or more individual names or a list of one or more data values, written $\exists C_1,\ldots,C_n$.
• A max-cardinality constraint with a non-negative integer $n$ followed (optionally) by either a class expression $C$ or a concrete-type-expression, written $\leqslant$$n.C$ ($\leqslant$$n.\top$ if the expression is omitted).
• A min-cardinality constraint with a non-negative integer $n$ followed (optionally) by either a class expression or a concrete type expression, written $\geqslant$$n.C$ ($\geqslant$$n.\top$ if the expression is omitted).
• A cardinality constraint with a non-negative integer $n$ followed (optionally) by either a class expression $C$ or a concrete type expression, written $=$$n.C$ ($=$$n.\top$ if the class expression is omitted).

In order to maintain the decidability of the language, cardinality constraints can only be applied to simple slots. A simple slot is one that is neither transitive nor has any transitive subslots. However, as the transitivity of a slot can be inferred (e.g., from the fact that the inverse of the slot is a transitive slot), simple slot is defined in terms of the translation into $\mathcal{SHIQ}(d)$: a slot $\textsf{\textsl{SN}}$ in an ontology $\mathcal{O}$ is a simple slot iff $\sigma(\ensuremath{\textsf{\textsl{SN}}})$ is a simple role in the $\mathcal{SHIQ}(d)$ terminology $\sigma(\ensuremath{\mathcal{O}}\xspace )$.