A Note on Encoding Inverse Roles and Functional Restrictions in ALC Knowledge Bases

In this paper we show that it is possible to eliminate inverse roles and functional restrictions from ALCFI knowledge bases, while preserving the soundness and completeness of inference. Specifically, we present two polynomial encodings, the first from ALCFI knowledge bases intoALCI ones, and the second fromALCI knowledge bases into ALC ones. These encodings eliminate functional restrictions and inverse roles respectively, but add enough information so as not to destroy the meaning of concepts in the original knowledge base with respect to the reasoning tasks (in particular we will focus on logical implication). The encodings presented here are derived from those in [De Giacomo and Lenzerini, 1994] and in [De Giacomo, 1996] (the latter in the context of Propositional Dynamic Logics) for much more expressive description logics, in which complex roles formed as regular expressions of atomic ones (including the reflexive-transitive closure) are allowed. Observe that, if we apply directly the encodings in [De Giacomo and Lenzerini, 1994; De Giacomo, 1996] to ALCFI knowledge bases, reflexive-transitive closure would be introduced to internalize axioms, and hence it would appear in the syntactic closure as well. As a consequence the resulting formula would not be expressible as an ALCFI knowledge base. However it can be shown that the parts not expressible as ALCFI assertions can be dropped without influencing the reasoning tasks. The encodings presented here are devised by making use of this result. Encoding inverse roles and functional restrictions in ALC knowledge bases, on the one hand, is of practical interest, since it allows for basing the “core inference procedures” for logical implication in ALCFI on the inference procedures for logical implication inALC, which are typically more efficient (e.g. constraint systems [Buchheit et al., 1993]) and for which implemented systems are already available (e.g. FACT [Horrocks, 1997]). On the other hand, such encodings are a simple illustration of a general technique for deriving reasoning procedures for expressive logics based on a (possibly polynomial) encoding of such logics into simpler ones. Intuitively, the technique is based on two main steps. Let the “Source Logic” be SL and the “Target Logic” be TL: 1. Identify a finite set of assertion schemas in the language of TL capturing those characteristics that distinguish SL from TL.