Hybrid Unification in the Description Logic EL

The description logic EL belongs to the family of logic-based knowledge representation formalisms. It allows a user to define concepts with the help of concept names (Nc), role names (Nr) and constructors: conjunction (u), existential restriction (∃r.C for r ∈ Nr and a concept C) and top constructor (>). Unification in Description Logics has been introduced in [6]. A unification problem in such logic is defined as a set of subsumptions between concepts which contain occurrences of a distinct set of concept names (called variables) and asks for definitions of these concept names, which would make the subsumptions valid. Unification in EL corresponds to unification modulo semilattices with monotone operators [5]. In [4], we were able to show that unification in EL is NP-complete. The problem is how to extend the unification in EL to such unification with a background ontology in the form of a set of definitions of some concept names occuring in the unification problem, or more generally in the form of additional statements about concept inclusions. If the background ontology is just a set of non-cyclic definitions, unification in EL is NP-complete [5]. If the background ontology satisfies some cycle restriction, it is still NP-complete [2]. At the moment it is not known what is the status of the unification problem in EL with a background ontology in the general case. In this paper, instead of restricting the background ontology, we allow cyclic definitions to be used as unifiers. Moreover, we interpret these definitions in a greatest fixpoint semantics, while the background ontology is still interpreted in the usual descriptive semantics. We show that if the concept of unification in EL is modified in this way, such unification is NP-complete. Detailed proofs and examples can be found in [3].