The mapping function $\sigma(\cdot)$

We can now define how the function $\sigma(\cdot)$ maps OIL axioms and class definitions into (sets of) $\mathcal{SHIQ}(d)$ axioms. The definition is given in Figures 2 and 3, where $\textsf{CN}$ is a class name (or a $\mathcal{SHIQ}(d)$ concept name), $\textsf{\textsl{SN}}$ is a slot name (or $\mathcal{SHIQ}(d)$ role name), $C$ (possibly subscripted) is a class expression, $D$ (possibly subscripted) is a class or concrete type expression, $E$ is a class expression or a class description (super-classes plus slot constraints), $A$ (possibly subscripted) is a slot constraint, $a_i$ is a constraint (on a slot), $i$ is an OIL individual, $P_i$ is the $\mathcal{SHIQ}(d)$ primitive concept used to represent the OIL individual $i$, $d$ is a concrete data value (an integer or a string), $n$ is a non-negative integer and $p_d$ is a unary predicate (i.e., $p \in \{\geqslant, \leqslant, >, <\}$ and $d$ is a concrete data value).

Figure: Translation of OIL axioms into $\mathcal{SHIQ}(d)$

Figure: Translation of OIL class definitions into $\mathcal{SHIQ}(d)$

In addition, the set of disjointness axioms $\mathcal{D}$ is defined as:

$$\bigcup_{j=1}^{n-1}\bigcup_{k=j+1}^{n}\{P_j \sqsubseteq \neg P_k\}$$

where $i_i,\ldots,i_n$ are the individuals used in $\mathcal{O}$ and $P_i$ is the $\mathcal{SHIQ}(d)$ primitive concept used to represent $i$.