Deciding Unifiability and Computing Local Unifiers in the Description Logic EL without Top Constructor

Abstract

Unification in description logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive description logic $\mathcal{EL}$ is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using $\mathcal{EL}$. On the other hand, unification in $\mathcal{EL}$ has been shown to be NP-complete and, thus, of considerably lower complexity than unification in other description logics of similarly restricted expressive power.

However, $\mathcal{EL}$ allows the use of the top concept ($\top$), which represents the whole interpretation domain, whereas the large medical ontology SNOMED CT makes no use of this feature. Surprisingly, removing the top concept from $\mathcal{EL}$ makes the unification problem considerably harder. More precisely, we will show that unification in $\mathcal{EL}$ without the top concept is PSPACE-complete. In addition to the decision problem, we also consider the problem of actually computing ${\mathcal{EL}}^{-\top }$-unifiers.