Slot definitions

A slot definition is a pair $\langle \ensuremath{\textsf{\textsl{SN}}},X \rangle$, where $\textsf{\textsl{SN}}$ is a slot name and $X$ is a slot description. A slot description $X$ consists of an optional subslot-of component, itself a list of one or more slot names $\ensuremath{\textsf{\textsl{RN}}}_1,\ldots,\ensuremath{\textsf{\textsl{RN}}}_n$, followed by a list of zero or more global slot constraints (e.g., inverse) $S_1,\ldots,S_m$. We will write such a slot definition as:

$$\ensuremath{\textsf{\textsl{SN}}}[\ensuremath{\textsf{\textsl{RN}}}_1,\ldots,\ensuremath{\textsf{\textsl{RN}}}_n,S_1,\ldots,S_m]$$

Each global constraint $S_i$ on $\textsf{\textsl{SN}}$ can be either:

• A domain constraint with a list of one or more class-expressions, written $\downarrow[C_1,\ldots,C_n]$.
• A range constraint with a list of one or more class-expressions, written $\uparrow[C_1,\ldots,C_n]$.
• An inverse constraint with a slot name $\textsf{\textsl{RN}}$, written $^-\ensuremath{\textsf{\textsl{RN}}}$.
• A properties constraint with a list of one or more properties, written $[P_1,\ldots,P_n]$. Valid properties are transitive, written $+$, symmetrical, written $\leftrightarrow$ and functional, written $\uparrow$.

We can now define how the function $\sigma(\cdot)$ maps an OIL slot definition into a set of $\mathcal{SHIQ}$ axioms. The definition is given in Figure 4, where $\textsf{\textsl{RN}}$ and $\textsf{\textsl{SN}}$ are slot names (or $\mathcal{SHIQ}$ role names), $C_i$ is a class expression, $S_i$ is a global slot constraint and $P_i$ is a property.

Figure: Translation of OIL slot definitions into $\mathcal{SHIQ}$