Closed Predicates in Description Logics: Results on Combined Complexity
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Some applications of Description Logic (DL) ontologies combine complete information (e.g., stemming from relational databases) with incomplete, open-world knowledge. Several research efforts in the last years have advocated closed predicates, which are predicates whose extension is interpreted as complete, as a suitable way to leverage partial completeness within the standard open-world semantics of DLs. These works have also studied the data complexity of query answering in the presence of closed predicates, which is generally intractable. In this paper we contribute to the understanding the combined complexity of the problem, by establishing tight complexity results for a range of DLs and query answering problems. In summary, our results show that consistency testing and instance query answering in the presence of closed predicates are feasible in NP even for rich dialects of the DL-Lite family; this is the lowest complexity that could be expected. For EL, in contrast, they are EXPTIME-complete, thus as hard as for ALC and some of its extensions. If unions of conjunctive queries (UCQs) are considered, the picture is even bleaker: we can show 2EXPTIME-hardness even for DL-Lite_R and EL. This is in sharp contrast to the NP-upper bound in the standard setting without closed predicates, and coincides with known upper bounds for much richer DLs. We note that our results imply 2EXPTIME-hardness of query answering in ALCO for the standard setting, where all predicates are interpreted under the open-world semantics. This singles out nominals as a previously unidentified source of complexity when answering queries over expressive DLs. Despite these negative results, we can still identify several useful classes of queries for which the increase in hardness is not as drastic, and the combined complexity of query answering remains between NP and coNEXPTIME.