Relational expressive power of constraint query languages

Abstract

The expressive power of first-order query languages with several classes of equality and inequality constraints is studied in this paper. We settle the conjecture that recursive queries such as parity test and transitive closure cannot be expressed in the relational calculus augmented with polynomial inequality constraints over the reals. Furthermore, noting that relational queries exhibit several forms of genericity, we establish a number of collapse results of the following form: The class of generic Boolean queries expressible in the relational calculus augmented with a given class of constraints coincides with the class of queries expressible in the relational calculus (with or without an order relation). We prove such results for both the natural and active-domain semantics. As a consequence, the relational calculus augmented with polynomial inequalities expresses the same classes of generic Boolean queries under both the natural and active-domain semantics.In the course of proving these results for the active-domain semantics, we establish Ramsey-type theorems saying that any query involving certain kinds of constraints coincides with a constraint-free query on databases whose elements come from a certain infinite subset of the domain. To prove the collapse results for the natural semantics, we make use of techniques from nonstandard analysis and from the model theory of ordered structures.

The Natural Order-Generic Collapse for ω-Representable Databases over the Rational and the Real Ordered Group

“…(This would then fully generalize the results of Belegradek et al [1]. ) We also want to mention a potential application concerning topological queries: Kuijpers and Van den Bussche [7] used the theorem of Benedikt et al [2] to obtain a collapse result for topological first-order definable queries. One step of their proof was to encode spatial databases (of a certain kind) by finite databases, to which the result of [2] can be applied.…”

“…In this setting Benedikt et al [2] have obtained a strong collapse result: Firstorder logic has the natural order-generic collapse for finite databases over ominimal structures. This means that if the universe U together with the additional predicates, has a certain property called o-minimality, then for every order-generic first-order formula ϕ which uses the additional predicates, there is a formula with linear ordering alone which is equivalent to ϕ on all finite databases.…”

“…) We also want to mention a potential application concerning topological queries: Kuijpers and Van den Bussche [7] used the theorem of Benedikt et al [2] to obtain a collapse result for topological first-order definable queries. One step of their proof was to encode spatial databases (of a certain kind) by finite databases, to which the result of [2] can be applied. Here the question arises whether there is an interesting class of spatial databases that can be encoded by ω-representable (but not by finite) databases in such a way that our main theorem helps to obtain some collapse result for topological queries.…”

Expressing the box cone radius in the relational calculus with real polynomial constraints

“…Proof of Claim. We prove the claim for 2 , the other case being subcases. We compute the differential d x ( f | Z ) as follows: Locally around x, we may assume that the projection on the first k coordinates : Hence, the product …”

“…Indeed, the expressive power of FO in expressing topological properties is rather limited. For example, topological connectivity is not expressible in FO [2]. Therefore, recursive extensions of FO have been studied in order to express more queries [12], [15], [17], [14], [7], [6].…”