Extended order-generic queries

Abstract

We consider relational databases organized over an ordered domain with some additional relations — a typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving “permutations” — such queries are called order-generic. It has recently been discovered that for some domains order-generic FO queries fail to express more than pure order queries. For example, every order-generic FO query over rational numbers with + can be rewritten without +. For some other domains, however, this is not the case. We provide very general conditions on the FO theory of the domain that ensure the collapse of order-generic extended FO queries to pure order queries over this domain: the Pseudo-finite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of domains satisfying the Isolation Property, the so-called quasi-o-minimal domains. This class includes all the o-minimal domains, but also the ordered group of integer numbers and the ordered semigroup of natural numbers, and some other domains. An important difference of this paper from the recent series of related papers is that we generalize all the notions to the case of finitely representable database states — as opposed to finite states — and develop a general lifting technique that, essentially, allows us to extend any result of the kind we are interested in, from finite to finitely representable states. We show, however, that these results cannot be transfered to arbitrary infinite states.