Dichotomies in Ontology-Mediated Querying with the Guarded Fragment

We study the complexity of ontology-mediated querying when ontologies are formulated in the guarded fragment of first-order logic (GF). Our general aim is to classify the data complexity on the level of ontologies where query evaluation w.r.t. an ontology O is considered to be in PTime if all (unions of conjunctive) queries can be evaluated in PTime w.r.t. O and coNP-hard if at least one query is coNP-hard w.r.t. O. We identify several large and relevant fragments of GF that enjoy a dichotomy between PTime and coNP, some of them additionally admitting a form of counting. In fact, almost all ontologies in the BioPortal repository fall into these fragments or can easily be rewritten to do so. We then establish a variation of Ladner's Theorem on the existence of NP-intermediate problems and use this result to show that for other fragments, there is provably no such dichotomy. Again for other fragments (such as full GF), establishing a dichotomy implies the Feder-Vardi conjecture on the complexity of constraint satisfaction problems. We also link these results to Datalog-rewritability and study the decidability of whether a given ontology enjoys PTime query evaluation, presenting both positive and negative results

Complexity of Approximate Query Answering under Inconsistency in Datalog+/-

This is the author accepted manuscript. The final version is available from the publisher via the link in this recordSeveral semantics have been proposed to query inconsistent ontological knowledge bases, including the intersection of repairs and the intersection of closed repairs as two approximate inconsistency-tolerant semantics. In this paper, we analyze the complexity of conjunctive query answering under these two semantics for a wide range of Datalog± languages. We consider both the standard setting, where errors may only be in the database, and the generalized setting, where also the rules of a Datalog± knowledge base may be erroneous.This work was supported by The Alan Turing Institute under the UK EPSRC grant EP/N510129/1, and by the EPSRC grants EP/R013667/1, EP/L012138/1, and EP/M025268/1

Complexity of Inconsistency-Tolerant Query Answering in Datalog+/- under Cardinality-Based Repairs

This is the author accepted manuscript. The final version is available from Association for the Advancement of Artificial Intelligence (AAAI) via the link in this recordQuerying inconsistent ontological knowledge bases is an important problem in practice, for which several inconsistencytolerant query answering semantics have been proposed, including query answering relative to all repairs, relative to the intersection of repairs, and relative to the intersection of closed repairs. In these semantics, one assumes that the input database is erroneous, and the notion of repair describes a maximally consistent subset of the input database, where different notions of maximality (such as subset and cardinality maximality) are considered. In this paper, we give a precise picture of the computational complexity of inconsistencytolerant (Boolean conjunctive) query answering in a wide range of Datalog± languages under the cardinality-based versions of the above three repair semantics.This work was supported by the Alan Turing Institute under the UK EPSRC grant EP/N510129/1, and by the EPSRC grants EP/R013667/1, EP/L012138/1, and EP/M025268/1

A data complexity and rewritability tetrachotomy of ontology-mediated queries with a covering axiom

Aiming to understand the data complexity of answering conjunctive queries mediated by an axiom stating that a class is covered by the union of two other classes, we show that deciding their first-order rewritability is PSPACE-hard and obtain a number of sufficient conditions for membership in AC0, L, NL, and P. Our main result is a complete syntactic AC0/NL/P/CONP tetrachotomy of path queries under the assumption that the covering classes are disjoint

Dichotomies in Ontology-Mediated Querying with the Guarded Fragment

We study ontology-mediated querying in the case where ontologies are formulated in the guarded fragment of first-order logic (GF) or extensions thereof with counting and where the actual queries are (unions of) conjunctive queries. Our aim is to classify the data complexity and Datalog rewritability of query evaluation depending on the ontology O, where query evaluation w.r.t. O is in PTime (resp. Datalog rewritable) if all queries can be evaluated in PTime w.r.t. O (resp. rewritten into Datalog under O), and coNP-hard if at least one query is coNP-hard w.r.t. O. We identify several fragments of GF that enjoy a dichotomy between Datalog-rewritability (which implies PTime) and coNP-hardness as well as several other fragments that enjoy a dichotomy between PTime and coNP-hardness, but for which PTime does not imply Datalog-rewritability. For the latter, we establish and exploit a connection to constraint satisfaction problems. We also identify fragments for which there is no dichotomy between PTime and coNP. To prove this, we establish a non-trivial variation of Ladner’s theorem on the existence of NP-intermediate problems. Finally, we study the decidability of whether a given ontology enjoys PTime query evaluation, presenting both positive and negative results, depending on the fragment

A tetrachotomy of ontology-mediated queries with a covering axiom

Our concern is the problem of efficiently determining the data complexity of answering queries mediated by descrip- tion logic ontologies and constructing their optimal rewritings to standard database queries. Originated in ontology- based data access and datalog optimisation, this problem is known to be computationally very complex in general, with no explicit syntactic characterisations available. In this article, aiming to understand the fundamental roots of this difficulty, we strip the problem to the bare bones and focus on Boolean conjunctive queries mediated by a simple cov- ering axiom stating that one class is covered by the union of two other classes. We show that, on the one hand, these rudimentary ontology-mediated queries, called disjunctive sirups (or d-sirups), capture many features and difficulties of the general case. For example, answering d-sirups is Π2p-complete for combined complexity and can be in AC0 or L-, NL-, P-, or coNP-complete for data complexity (with the problem of recognising FO-rewritability of d-sirups be- ing 2ExpTime-hard); some d-sirups only have exponential-size resolution proofs, some only double-exponential-size positive existential FO-rewritings and single-exponential-size nonrecursive datalog rewritings. On the other hand, we prove a few partial sufficient and necessary conditions of FO- and (symmetric/linear-) datalog rewritability of d- sirups. Our main technical result is a complete and transparent syntactic AC0 / NL / P / coNP tetrachotomy of d-sirups with disjoint covering classes and a path-shaped Boolean conjunctive query. To obtain this tetrachotomy, we develop new techniques for establishing P- and coNP-hardness of answering non-Horn ontology-mediated queries as well as showing that they can be answered in NL

A tetrachotomy of ontology-mediated queries with a covering axiom

Our concern is the problem of efficiently determining the data complexity of answering queries mediated by descrip- tion logic ontologies and constructing their optimal rewritings to standard database queries. Originated in ontology- based data access and datalog optimisation, this problem is known to be computationally very complex in general, with no explicit syntactic characterisations available. In this article, aiming to understand the fundamental roots of this difficulty, we strip the problem to the bare bones and focus on Boolean conjunctive queries mediated by a simple cov- ering axiom stating that one class is covered by the union of two other classes. We show that, on the one hand, these rudimentary ontology-mediated queries, called disjunctive sirups (or d-sirups), capture many features and difficulties of the general case. For example, answering d-sirups is Π2p-complete for combined complexity and can be in AC0 or L-, NL-, P-, or coNP-complete for data complexity (with the problem of recognising FO-rewritability of d-sirups be- ing 2ExpTime-hard); some d-sirups only have exponential-size resolution proofs, some only double-exponential-size positive existential FO-rewritings and single-exponential-size nonrecursive datalog rewritings. On the other hand, we prove a few partial sufficient and necessary conditions of FO- and (symmetric/linear-) datalog rewritability of d- sirups. Our main technical result is a complete and transparent syntactic AC0 / NL / P / coNP tetrachotomy of d-sirups with disjoint covering classes and a path-shaped Boolean conjunctive query. To obtain this tetrachotomy, we develop new techniques for establishing P- and coNP-hardness of answering non-Horn ontology-mediated queries as well as showing that they can be answered in NL