In this paper, we study the data complexity of querying inconsistent weighted description logic (DL) knowledge bases under recently-introduced cost-based semantics. In a nutshell, the idea is to assign each interpretation a cost based upon the weights of the violated axioms and assertions, and certain and possible query answers are determined by considering all (resp. some) interpretations having optimal or bounded cost. Whereas the initial study of cost-based semantics focused on DLs between EL_bot and ALCO, we consider DLs that may contain inverse roles and role inclusions, thus covering prominent DL-Lite dialects. Our data complexity analysis goes significantly beyond existing results by sharpening several lower bounds and pinpointing the precise complexity of optimal-cost certain answer semantics (no non-trivial upper bound was known). Moreover, while all existing results show the intractability of cost-based semantics, our most challenging and surprising result establishes that if we consider DL-Lite^H_bool ontologies and a fixed cost bound, certain answers for instance queries and possible answers for conjunctive queries can be computed using first-order rewriting and thus enjoy the lowest possible data complexity (AC0).

Recent works have explored the use of counting queries coupled with Description Logic ontologies. The answer to such a query in a model of a knowledge base is either an integer or infinity, and its spectrum is the set of its answers over all models. While it is unclear how to compute and manipulate such a set in general, we identify a class of counting queries whose spectra can be effectively represented. Focusing on atomic counting queries, we pinpoint the possible shapes of a spectrum over ALCIF ontologies: they are essentially the subsets of N and infinity closed under addition. For most sublogics of ALCIF, we show that possible spectra enjoy simpler shapes, being [ m, infinity ] or variations thereof. To obtain our results, we refine constructions used for finite model reasoning and notably rely on a cycle-reversion technique for the Horn fragment of ALCIF. We also study the data complexity of computing the proposed effective representation and establish the FP^NP[log]-completeness of this task under several settings.

Reasoning with minimal models has always been at the core of many knowledge representation techniques, but we still have only a limited understanding of this problem in Description Logics (DLs). Minimization of some selected predicates---letting the remaining predicates vary or be fixed, as proposed in circumscription---has been explored and exhibits high complexity. The case of `pure' minimal models, where the extension of all predicates must be minimal, has remained largely uncharted. We address this problem in popular DLs and obtain surprisingly negative results: concept satisfiability in minimal models is undecidable already for EL. This undecidability also extends to a very restricted fragment of tuple-generating dependencies. To regain decidability, we impose acyclicity conditions on the TBox that bring the worst-case complexity below double exponential time and allow us to establish a connection with the recently studied pointwise circumscription; we also derive results in data complexity. We conclude with a brief excursion to the DL-Lite family, where a positive result was known for DL-Lite_core, but our investigation establishes ExpSpace-hardness already for its extension DL-Lite_horn.

We study extensions of expressive decidable fragments of first-order logic with circumscription, considering in particular the two-variable fragment FO^2, its extension C^2 with counting quantifiers, and the guarded fragment GF. We prove that if only unary predicates are minimized (or fixed) during circumscription, then decidability of logical consequence is preserved. For FO^2 the complexity increases from NExp to NExp^NP-complete, for GF it (remarkably!) increases from 2Exp to Tower-complete, and for C^2 it remains open. We also consider querying circumscribed knowledge bases whose ontology is a GF sentence, showing that the problem is decidable for unions of conjunctive queries, Tower-complete in combined complexity, and elementary in data complexity. Already for atomic queries and ontologies that are sets of guarded existential rules, however, for every k > 0 there is an ontology and query that are k-Exp-hard in data complexity.

Circumscription is one of the main approaches for defining non-monotonic description logics (DLs) and the decidability and complexity of traditional reasoning tasks, such as satisfiability of circumscribed DL knowledge bases (KBs) are well understood. For evaluating conjunctive queries (CQs) and unions thereof (UCQs), in contrast, not even decidability had been established. In this paper, we prove decidability of (U)CQ evaluation on circumscribed DL KBs and obtain a rather complete picture of both the combined complexity and the data complexity, for DLs ranging from ALCHIO via EL to various versions of DL-Lite. We also study the much simpler atomic queries (AQs).

Ontology-mediated query answering (OMQA) employs structured knowledge and automated reasoning in order to facilitate access to incomplete and possibly heterogeneous data. While most research on OMQA adopts (unions of) conjunctive queries as the query language, there has been recent interest in handling queries that involve counting. In this paper, we advance this line of research by investigating cardinality queries (which correspond to Boolean atomic counting queries) coupled with DL-Lite ontologies. Despite its apparent simplicity, we show that such an OMQA setting gives rise to rich and complex behaviour. While we prove that cardinality query answering is tractable (TC0) in data complexity when the ontology is formulated in DL-Lite-core, the problem becomes coNP-hard as soon as role inclusions are allowed. For DL-Lite-pos-H (which allows only positive axioms), we establish a P-coNP dichotomy and pinpoint the TC0 cases; for DL-Lite-core-H (allowing also negative axioms), we identify new sources of coNP complexity and also exhibit L-complete cases. Interestingly, and in contrast to related tractability results, we observe that the canonical model may not give the optimal count value in the tractable cases, which led us to develop an entirely new approach based upon exploring a space of strategies to determine the minimum possible number of query matches.

Ontology-mediated query answering (OMQA) is a promising approach to data access and integration that has been actively studied in the knowledge representation and database communities for more than a decade. The vast majority of work on OMQA focuses on conjunctive queries, whereas more expressive queries that feature counting or other forms of aggregation remain largely unexplored. In this paper, we introduce a general form of counting query, relate it to previous proposals, and study the complexity of answering such queries in the presence of DL-Lite ontologies. As it follows from existing work that query answering is intractable and often of high complexity, we consider some practically relevant restrictions, for which we establish improved complexity bounds.