We study the interaction of views, queries, and background knowledge in the form of existential rules. The motivating questions concern monotonic determinacy of a query using views w.r.t. rules, which refers to the ability to recover the query answer from the views via a monotone function. We study the decidability of monotonic determinacy, and compare with variations that require the “recovery function” to be in a well-known monotone query language, such as conjunctive queries or Datalog. Surprisingly, we find that even in the presence of basic existential rules, the borderline between well-behaved and badly-behaved answerability differs radically from the unconstrained case. In order to understand this boundary, we require new results concerning entailment problems involving views and rules.

A query Q is monotonically determined over a set of views V if Q can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for conjunctive query views and queries. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.

Our concern is the problem of efficiently determining the data complexity of answering queries mediated by description 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 covering 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 or L-, NL-, P-, or coNP-complete for data complexity (with the problem of recognising FO-rewritability of d-sirups being 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 /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.

We show that deciding boundedness (aka FO-rewritability) of mon­adic single rule datalog programs (sirups) is 2\Exp-hard, which matches the upper bound known since 1988 and finally settles a long-standing open problem. We obtain this result as a byproduct of an attempt to classify monadic 'disjunctive sirups'---Boolean conjunctive queries $\q$ with unary and binary predicates mediated by a disjunctive rule $T(x) łor F(x) łeftarrow A(x)$---according to the data complexity of their evaluation. Apart from establishing that deciding FO-rewritability of disjunctive sirups with a dag-shaped $\q$ is also 2\Exp-hard, we make substantial progress towards obtaining a complete FO/Ł-hardness dichotomy of disjunctive sirups with ditree-shaped $\q$.

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.

Abstract Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic.

We give solutions to two fundamental computational problems in ontology-based data access with the W3C standard ontology language OWL 2 QL : the succinctness problem for first-order rewritings of ontology-mediated queries (OMQs) and the complexity problem for OMQ answering. We classify OMQs according to the shape of their conjunctive queries (treewidth, the number of leaves) and the existential depth of their ontologies. For each of these classes, we determine the combined complexity of OMQ answering and whether all OMQs in the class have polynomial-size first-order, positive existential, and nonrecursive datalog rewritings. We obtain the succinctness results using hypergraph programs, a new computational model for Boolean functions, which makes it possible to connect the size of OMQ rewritings and circuit complexity.

Our concern is the overhead of answering OWL 2 QL ontology-mediated queries (OMQs) in ontology-based data access compared to evaluating their underlying tree-shaped and, more generally, bounded treewidth conjunctive queries (CQs). We show that OMQs with bounded depth ontologies have nonrecursive datalog (NDL) rewritings that can be constructed and evaluated in LOGCFL for combined complexity, and even in NL if their CQs are tree-shaped with a bounded number of leaves. Thus, such OMQs incur no overhead in complexity-theoretic terms. For OMQs with arbitrary ontologies and bounded-leaf tree-shaped CQs, NDL-rewritings are constructed and evaluated in LOGCFL. We experimentally demonstrate feasibility and scalability of our rewritings compared to previously proposed NDL-rewritings. On the negative side, we prove that answering OMQs with tree-shaped CQs is not fixed-parameter tractable if the ontology depth or the number of leaves in the CQs is regarded as the parameter, and that answering OMQs with a fixed ontology (of infinite depth) is NP-complete for tree-shaped CQs and LOGCFL-complete for bounded-leaf CQs.

This paper investigates the impact of query topology on the difficulty of answering conjunctive queries in the presence of OWL 2 QL ontologies. Our first contribution is to clarify the worst-case size of positive existential (PE), non-recursive Data log (NDL), and first-order (FO) rewritings for various classes of tree-like conjunctive queries, ranging from linear queries to bounded tree width queries. Perhaps our most surprising result is a super polynomial lower bound on the size of PE-rewritings that holds already for linear queries and ontologies of depth 2. More positively, we show that polynomial-size NDL-rewritings always exist for tree-shaped queries with a bounded number of leaves (and arbitrary ontologies), and for bounded tree width queries paired with bounded depth ontologies. For FO-rewritings, we equate the existence of polysize rewritings with well-known problems in Boolean circuit complexity. As our second contribution, we analyze the computational complexity of query answering and establish tractability results (either NL-or LOGCFL-completeness) for a range of query-ontology pairs. Combining our new results with those from the literature yields a complete picture of the succinctness and complexity landscapes for the considered classes of queries and ontologies.

Stanislav Kikot; Relation Algebras by Games, by Robin Hirsch and Ian Hodkinson, Journal of Logic and Computation, Volume 20, Issue 2, 1 April 2010, Pages 6