next up previous
Next: Slot definitions Up: OIL-Lite Previous: Axioms

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)$
\begin{figure}\begin{displaymath}
\ \begin{array}{r@{\quad=\quad}l}
\ \sigma(C \...
... \sigma(C_i \doteq C_{i+1}) \\
\ \end{array}\ \end{displaymath}\ \ \end{figure}

Figure: Translation of OIL class definitions into $ \mathcal{SHIQ}(d)$
\begin{figure}\begin{displaymath}
\ \begin{array}{r@{\quad=\quad}l}
\ \sigma([C_...
..._{d}$}})}}\\
\ \sigma(p_d) & p_d
\ \end{array}\ \end{displaymath}\ \end{figure}

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

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

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$.


next up previous
Next: Slot definitions Up: OIL-Lite Previous: Axioms
Ian Horrocks 2000-09-10