We give a solution to the succinctness problem for the size of first-order rewritings of conjunctive queries in ontologybased data access with ontology languages such as OWL 2 QL, linear Datalog ± and sticky Datalog ± . We show that positive existential and nonrecursive datalog rewritings, which do not use extra non-logical symbols (except for intensional predicates in the case of datalog rewritings), suffer an exponential blowup in the worst case, while first-order rewritings can grow superpolynomially unless NP ⊆ P/poly. We also prove that nonrecursive datalog rewritings are in general exponentially more succinct than positive existential rewritings, while first-order rewritings can be superpolynomially more succinct than positive existential rewritings. On the other hand, we construct polynomial-size positive existential and nonrecursive datalog rewritings under the assumption that any data instance contains two fixed constants.

Abstract. We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of firstorder and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove similar results for the size of rewritings that do not use non-signature constants. For example, we show that, in the worst case, positive existential and nonrecursive Datalog rewritings are exponentially longer than the original queries; nonrecursive Datalog rewritings are in general exponentially more succinct than positive existential rewritings; while first-order rewritings can be superpolynomially more succinct than positive existential rewritings.

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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 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, even in NL if their CQs are tree-shaped with a bounded number of leaves, and so incur no overhead in complexity-theoretic terms. For OMQs with arbitrary ontologies and bounded-leaf CQs, NDL-rewritings are constructed and evaluated in LOGCFL. We show experimentally feasibility and scalability of our rewritings compared to previously proposed NDL-rewritings. On the negative side, we prove that answering OMQs with treeshaped 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 and LOGCFL for bounded-leaf CQs.

A tropical (or min-plus) semiring is a set Z (or Z ∪ {∞}) endowed with two operations: ⊕, which is just usual minimum, and ⊙, which is usual addition. In tropical algebra the vector x is a solution to a polynomial, where g i (x)'s are tropical monomials, if the minimum in min i (g i (x)) is attained at least twice. In min-plus algebra solutions of systems of equations of the formIn this paper we consider computational problems related to tropical linear system. We show that the solvability problem (both over Z and Z∪{∞}) and the problem of deciding the equivalence of two linear systems (both over Z and Z ∪ {∞}) are equivalent under polynomialtime reduction to mean payoff games and are also equivalent to analogous problems in min-plus algebra. In particular, all these problems belong to NP ∩ coNP. Thus we provide a tight connection of computational aspects of tropical linear algebra with mean payoff games and min-plus linear algebra. On the other hand we show that computing the dimension of the solution space of a tropical linear system and of a min-plus linear system are NP-complete.We also extend some of our results to the systems of min-plus linear inequalities.

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We investigate the succinctness problem for conjunctive query rewritings over OWL 2 QL ontologies of depth 1 and 2 by means of hypergraph programs computing Boolean functions. Both positive and negative results are obtained. We show that, over ontologies of depth 1, conjunctive queries have polynomial-size nonrecursive datalog rewritings; tree-shaped queries have polynomial positive existential rewritings; however, in the worst case, positive existential rewritings can be superpolynomial. Over ontologies of depth 2, positive existential and nonrecursive datalog rewritings of conjunctive queries can suffer an exponential blowup, while first-order rewritings can be superpolynomial unless NP ⊆ P/poly. We also analyse rewritings of tree-shaped queries over arbitrary ontologies and note that query entailment for such queries is fixed-parameter tractable.

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials.In this paper we prove a tropical Nullstellensatz and moreover we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field.On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems.