Finitely Representable Databases

Abstract

We study infinite but finitely representable databases based on constraints, motivated by new database applications such as those involving spatio-temporal information. We introduce a general definition of finite representation and define the concept of a query as a generalization of a query over relational databases. We investigate the theory of finitely representable models and prove that it differs from both classical model theory and finite model theory. In particular, we show the failure of most of the well-known theorems of logic (compactness, completeness, etc.). An important consequence is that properties such as query satisfiability and containment are undecidable. We illustrate the use of Ehrenfeucht–Fraı̈ssé games on the expressive power of query languages over finitely representable databases. As a case study, we focus on queries over dense order constraint databases. We consider in particular “order-generic” queries which are mappings closed under order-preserving bijections and topological queries, mappings closed under homeomorphisms. We prove that many interesting queries such as topological connectivity are not first-order definable with dense order constraints. We then consider an inflationary fixpoint query language, and prove that it captures exactly all PTIME order-generic queries. Finally, we give a rapid survey of recent results for more general contexts, such as polynomial constraints.