Variable independence for first-order definable constraints

Abstract

Whenever we have data represented by constraints (such as order, linear, polynomial, etc.), running time for many constraint processing algorithms can be considerably lowered if it is known that certain variables in those constraints are independent of each other. For example, when one deals with spatial and temporal databases given by constraints, the projection operation, which corresponds to quantifier elimination, is usually the costliest. Since the behavior of many quantifier elimination algorithms becomes worse as the dimension increases, eliminating certain variables from consideration helps speed up those algorithms.While these observations have been made in the literature, it remained unknown when the problem of testing if certain variables are independent is decidable, and how to efficiently construct a new representation of a constraint-set in which those variables do not appear together in the same atomic constraints. Here we answer this question. We first consider a general condition that gives us decidability of variable independence; this condition is stated in terms of model-theoretic properties of the structures corresponding to constraint classes. We then show that this condition covers the domains most relevant to spatial and temporal applications. For some of these domains, including linear and polynomial constraints over the reals, we provide a uniform decision procedure that gives us tractability as well. For those constraints, we also present a polynomial-time algorithm for producing nice constraint representations.