Combined Complexity of Answering Tree-like Queries in OWL 2 QL.

Introduction The OWL 2 QL ontology language [11], based upon the description logic DL-LiteR, is considered particularly well suited for applications involving large amounts of data. While the data complexity of querying OWL 2 QL knowledge bases is well understood, far less is known about combined complexity of conjunctive query (CQ) answering for restricted classes of conjunctive queries. By contrast, the combined complexity of CQ answering in the relational setting has been thoroughly investigated. In relational databases, it is well known that CQ answering is NP-complete in the general case. A seminal result by Yannakakis established the tractability of answering tree-shaped (aka acyclic) CQs [14], and this result was later extended to wider classes of queries, most notably to bounded treewidth CQs [5]. Gottlob et al. [6] pinpointed the precise complexity of answering tree-shaped and bounded treewidth CQs, showing both problems to be complete for the class LOGCFL of all languages logspace-reducible to context-free languages [13]. In the presence of arbitrary OWL 2 QL ontologies, the NP upper bound for arbitrary CQs continues to hold [4], but answering tree-shaped queries becomes NP-hard [8]. Interestingly, the latter problem was recently proven tractable in [3] for DL-Litecore (a slightly less expressive logic than OWL 2 QL), raising the hope that other restrictions might also yield tractability. This extended abstract summarizes our investigation [2] into the combined complexity of conjunctive query answering in OWL 2 QL for tree-shaped queries, their restriction to linear and bounded leaf queries and their generalization to bounded treewidth queries. Our complexity analysis reveals that all query-ontology combinations that have not already been shown NP-hard are in fact tractable. Specifically, in the case of bounded depth ontologies, we prove membership in LOGCFL for bounded treewidth queries (generalizing the result in [6]) and membership in NL for bounded leaf queries. We also show LOGCFL-completeness for linear and bounded leaf queries in the presence of arbitrary OWL 2 QL ontologies. This last result is the most interesting technically, as the upper and lower bounds rely on two different characterizations of the class LOGCFL.