We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. We prove that the added expressivity does not increase the complexity of satisfiability checking and other standard inference problems. However, reasoning in ALCSCC++ becomes undecidable if inverse roles are added or conjunctive query entailment is considered. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints are appropriately restricted. In this setting, query entailment can be shown to be EXPTIME-complete and hence not harder than reasoning in ALC.
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