Expressing Topological Connectivity of Spatial Databases

We consider two-dimensional spatial databases defined in terms of polynomial inequalities and focus on the potential of programming languages for such databases to express queries related to topological connectivity. It is known that the topological connectivity test is not first-order expressible. One approach to obtain a language in which connectivity queries can be expressed would be to extend FO+Poly with a generalized (or Lindstrom) quantifier expressing that two points belong to the same connected component of a given database. For the expression of topological connectivity, extensions of first-order languages with recursion have been studied (in analogy with the classical relational model). Two such languages are spatial Datalog and FO+Poly+WHILE. Although both languages allow the expression of non-terminating programs, their (proven for FO+Poly+WHILE and conjectured for spatial Datalog) computational completeness makes them interesting objects of study. Previously, spatial Datalog programs have been studied for more restrictive forms of connectivity (e.g., piece-wise linear connectivity) and these programs were proved to correctly test connectivity on restricted classes of spatial databases (e.g., linear databases) only. In this paper, we present a spatial Datalog program that correctly tests topological connectivity of arbitrary compact (i.e., closed and bounded) spatial databases. In particular, it is guaranteed to terminate on this class of databases. This program is based on a first-order description of a known topological property of spatial databases, namely that locally they are conical. We also give a very natural implementation of topological connectivity in FO+Poly+WHILE, that is based on a first-order implementation of the curve selection lemma, and that works correctly on arbitrary spatial databases inputs. Finally, we raise the question whether topological connectivity of arbitrary spatial databases can also be expressed in spatial Datalog.

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