Abstract
We introduce a new description logic that extends the well-known logic \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\) by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\). To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\), we are able to show that the complexity of reasoning in it is the same as in \(\mathcal {A}\mathcal {L}\mathcal {C}\mathcal {Q}\), both without and with TBoxes.
Partially supported by DFG within the Research Unit 1513 Hybris.
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Notes
- 1.
The name \(\mathcal {ALCSCC}\) for our new DL indicates that it extends the basic DL \(\mathcal {A}\mathcal {L}\mathcal {C}\) with set and cardinality constraints rather than just qualified number restrictions.
- 2.
In contrast to PA expressions, we do not have integer variables here and numerical constants must be non-negative.
- 3.
This is just like the QFBAPA formula \(\phi \) obtained from a Boolean valuation in our PSpace algorithm in the previous section.
- 4.
Note that [13] actually uses a different syntax for cardinality restrictions on role successors. To avoid having to introduce another syntax, we have translated this into our syntax. The constraint \( succ (r'\subseteq C)\) expresses the value restriction \(\forall r'.C\).
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Acknowledgment
The author thanks Viktor Kuncak for helpful discussions regarding the proof of Lemma 3.
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Baader, F. (2017). A New Description Logic with Set Constraints and Cardinality Constraints on Role Successors. In: Dixon, C., Finger, M. (eds) Frontiers of Combining Systems. FroCoS 2017. Lecture Notes in Computer Science(), vol 10483. Springer, Cham. https://doi.org/10.1007/978-3-319-66167-4_3
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