FroCoS 2019: Frontiers of Combining Systems pp 203-219

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

• Filippo De Bortoli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11715)

## 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.

## Notes

### Acknowledgment

The authors would like to thank Ulrike Baumann for helpful discussions regarding QFBAPA$$^\infty$$. We should also like to point out that we have learned about the results regarding QFBAPA$$_\infty$$ in [10] only a couple of days before the submission of the final version of this paper.

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