On the Expressive Power of Description Logics with Cardinality Constraints on Finite and Infinite Sets

Abstract

In recent work we have extended the description logic (DL) $\mathcal {ALC\!Q}$ by means of more expressive number restrictions using numerical and set constraints stated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA). It has been shown that reasoning in the resulting DL, called $\mathcal {ALCSCC}$, is PSpace-complete without a TBox and ExpTime-complete w.r.t. a general TBox. The semantics of $\mathcal {ALCSCC}$ is defined in terms of finitely branching interpretations, that is, interpretations where every element has only finitely many role successors. This condition was needed since QFBAPA considers only finite sets. In this paper, we first introduce a variant of $\mathcal {ALCSCC}$, called $\mathcal {ALCSCC} ^\infty$, in which we lift this requirement (inexpressible in first-order logic) and show that the complexity results for $\mathcal {ALCSCC}$ mentioned above are preserved. Nevertheless, like $\mathcal {ALCSCC}$, $\mathcal {ALCSCC} ^\infty$ is not a fragment of first-order logic. The main contribution of this paper is to give a characterization of the first-order fragment of $\mathcal {ALCSCC} ^\infty$. The most important tool used in the proof of this result is a notion of bisimulation that characterizes this fragment.

Partially supported by the German Research Foundation (DFG) within the Research Unit 1513 (Hybris) and the Research Training Group 1763 (QuantLA).