Dense order

Abstract: We consider infinite databases which admit a finite representation in terms of dense-order constraints. We study the complexity and the expressive power of various query languages over dense order constraint databases, allowing order, addition, recursion, or nested sets. We provide in particular an exact characterization of the class of dense order queries computable in PTIME (data complexity). We also prove that region and graph connectivity queries are not definable with linear constraints. We then investigate complex object models for constraint databases. Complex objects are fundamental to deal with pointsets as first-class citizens. We introduce an active domain semantics, and show that in terms of complexity and expressive power, the characterizations of the calculus for constraint complex objects are similar to the case of the classical complex object calculus.

Abstract: We consider the data complexity of various logics on two important classes of constraint databases: dense order and linear constraint databases. For dense order databases, we present a general result allowing us to lift results on logics capturing complexity classes from the class of finite ordered databases to dense order constraint databases. Considering linear constraints, we show that there is a significant gap between the data complexity of first-order queries on linear constraint databases over the real and the natural numbers. This is done by proving that for arbitrary high levels of the Presburger arithmetic there are complete first-order queries on databases over (N, <, +). The proof of the theorem demonstrates a simple argument for translating complexity results for prefix classes in logical theories to results on the complexity of query evaluation in constraint databases.

Abstract: Two infinite structures (sets together with operations and relations) hold our attention here: the trees together with operations of construction and the rational numbers together with the operations of addition and substraction and a linear dense order relation without endpoints. The object of this paper is the study of the evaluated trees, a structure mixing the two preceding ones. First of all, we establish a general theorem which gives a sufficient condition for the completeness of a first-order theory. This theorem uses a special quantifier, primarily asserting the existence of an infinity of individuals having a given first order property. The proof of the theorem is nothing other than the broad outline of a general algorithm which decides if a proposition or its negation is true in certain theories. We introduce then the theory TE of the evaluated trees and show its completeness using our theorem. From our proof it is possible to extract a general algorithm for solving quantified constraints in TE .