On the Expressive Power of Logics on Constraint Databases with Complex Objects

We extend the constraint data model to allow complex objects and study the expressive power of various query languages over this sort of constraint databases. The tools we use come in the form of collapse results which are well established in the context of first-order logic. We show that the natural-active collapse with a condition and the activegeneric collapse carry over to the second-order logic for structures with o-minimality property and any signature in the complex value relations. The expressiveness results for more powerful logics including monadic second-order logic, monadic second-order logic with fix-point operators, and fragments of second-order logic are investigated in the paper. We discuss the data complexity for second-order logics over constraint databases. The main results are that the complexity upper bounds for three theories, $\mathcal{M}\mathcal{S}\mathcal{O}$ + LIN, $\mathcal{M}\mathcal{S}\mathcal{O}$ + POLY, and Inflationary DATALOG ${}_{\mathrm{a}\mathrm{c}\mathrm{t}}^{\mathrm{c}\mathrm{v},\mathrm{\neg }}$ (SC, $\mathfrak{M}$) without powerset operator are ${\cup }_i{\sum }_i^{N{C}^1},\mathbf{N}\mathbf{C}\mathbf{H}={\cup }_i{\sum }_i^{NC}$, and AC⁰/poly, respectively. We also consider the problem of query closure property in the context of embedded finite models and constraint databases with complex objects and the issue of how to determine safe constraint queries.