On expressing topological connectivity in spatial datalog

Abstract

We consider two-dimensional spatial databases defined in terms of polynomial inequalities and investigate the expressibility of the topological connectivity query for these databases in spatial Datalog. In [10], a spatial Datalog program for piecewise linear connectivity was given and proved to correctly test the connectivity of linear spatial databases. In particular, the program was proved to terminate on these inputs. Here, we generalize this result and give a program that correctly tests connectivity of spatial databases definable by a quantifier-free formula in which at most quadratic polynomials appear. We also show that a further generalization of our approach to spatial databases that are only definable in terms of polynomials of higher degree is impossible. The class of spatial databases that can be defined by a quantifier-free formula in which at most quadratic polynomials appear is shown to be decidable. Finally, we give a number of possible other approaches to attack the problem of expressing the connectivity query for arbitrary two-dimensional spatial databases in spatial Datalog.