Obstacles on the Way to Spatial Reasoning with Description Logics: Undecidability of $\mathcal{ALC}_{RA}\ominus$
Michael Wessel
University of Hamburg, Computer Science Department, Vogt-Kölln-Str. 30, 22527 Hamburg, Germany
Abstract
This paper presents the new description logic $\mathcal{ALC}{RA}\ominus$. $\mathcal{ALC}{RA}\ominus$ combines the well-known standard description logic $\mathcal{ALC}$ with composition-based role axioms of the form $S \circ T \sqsubseteq R_1 \sqcup \cdots \sqcup R_n$. We argue that these axioms are nearly indispensable components in a description logic framework suitable for qualitative spatial reasoning tasks. An $\mathcal{ALC}{RA}\ominus$ spatial reasoning example is presented, and the relationships to other descriptions logics are discussed (namely $\mathcal{ALC}{R_A}$, $\mathcal{ALC}{R+}$, $\mathcal{ALC}\oplus$, $\mathcal{ALCH}{R+}$). Unfortunately, the satisfiability problem of this new logic is undecidable. Due to the high relevance of role axioms of the proposed form for all kinds of qualitative reasoning tasks, the undecidability of $\mathcal{ALC}{RA}\ominus$ is an important result.
1 Introduction and Motivation
Since the introduction of KL-ONE (see [2]), knowledge representation systems based on description logics (DLs) have been proven valuable tools in the field of formal knowledge representation. Description logic systems offer formally defined syntax and semantics, which enables the unambiguous specification of the services offered to users of these systems. In fact, many early knowledge representation systems and frameworks suffered from unclear semantics (e.g. see [23] for an overview and discussion). In many cases the underlying base description logic of a DL-based system can be seen as a subset of first order predicate logic (FOPL). In contrast to FOPL, decidability of diverse inference problems is usually guaranteed for description logics, for example, for the satisfiability problem