OIL semantics

The semantics of OIL rely on a translation into the description logic $\mathcal{SHIQ}$ [4] extended with concrete data types [2]: we will call this logic $\mathcal{SHIQ}(d)$. $\mathcal{SHIQ}(d)$ has a highly expressive concept language that is able to fully capture both the OIL-Lite and OIL-Standard languages, provided that OIL individuals are treated as ``syntactic sugar'' for disjoint primitive concepts, and we will define a translation function $\sigma(\cdot)$ that maps OIL ontologies into equivalent $\mathcal{SHIQ}(d)$ terminologies. This has the benefit that reasoning engines for OIL ontologies are (or will soon be) available: an existing $\mathcal{SHIQ}$ reasoner implemented in the FaCT system [3] can be used to reason with OIL ontologies not containing concrete data types, and this will soon be extended to a full $\mathcal{SHIQ}(d)$ reasoner. On the other hand, reasoning with ontologies containing individuals in class definitions seems to problematical, and the design of implementable algorithms is still an open problem.¹

The translation is quite straightforward and follows directly from the syntax and informal specification of OIL. An OIL ontology $\mathcal{O}$ consists of a list $d_1,\ldots,d_n$, where each $d_i$ is either a class definition, an axiom, a slot definition, an instantiation axiom or a relationship axiom. This list of definitions/axioms is translated into a $\mathcal{SHIQ}(d)$ terminology $\mathcal{T}$ (a set of axioms) as follows:

$$\sigma(d_1,\ldots,d_n) = \bigcup_{i=1}^{n} \sigma(d_i) \cup \mathcal{D}$$

where $\mathcal{D}$ is a set of axioms that enforces the disjointness of primitive concepts representing the individuals used in $\mathcal{O}$.

The syntactic correspondence between OIL and $\mathcal{SHIQ}(d)$ is summarised in Figure 1 and described in more detail in the following sections.²The translation function $\sigma(\cdot)$ is defined in Figures 2 to 4.

Figure: Syntactic correspondence between OIL and $\mathcal{SHIQ}(d)$