Expressing the box cone radius in the relational calculus with real polynomial constraints

Abstract We show that there is a query expressible in first-order logic over the reals that returns, on any given semi-algebraic set A, for every point, a radius around which A is conical in every small enough box. We obtain this result by combining results from differential topology and real algebraic geometry, with recent algorithmic results by Rannou.

[1]  Bart Kuijpers,et al.  Expressing Topological Connectivity of Spatial Databases , 1999, DBPL.

[2]  Gabriel M. Kuper,et al.  Tractable Recursion over Geometric Data , 1997, CP.

[3]  L. Dries,et al.  Geometric categories and o-minimal structures , 1996 .

[4]  Michael Benedikt,et al.  Reachability and connectivity queries in constraint databases , 2000, J. Comput. Syst. Sci..

[5]  Martin Grohe,et al.  On first-order topological queries , 2002, TOCL.

[6]  J. Risler,et al.  Real algebraic and semi-algebraic sets , 1990 .

[7]  Floris Geerts.  Linear Approximation of Semi-algebraic Spatial Databases Using Transitive Closure Logic, in Arbitrary Dimension , 2001, DBPL.

[8]  Lou van den Dries,et al.  Alfred Tarski's Elimination Theory for Real Closed Fields , 1988, J. Symb. Log..

[9]  Victor Vianu,et al.  Querying spatial databases via topological invariants , 1998, J. Comput. Syst. Sci..

[10]  M. Shiota.  Geometry of subanalytic and semialgebraic sets , 1997 .

[11]  E. Rannou,et al.  The Complexity of Stratification Computation , 1998, Discret. Comput. Geom..

[12]  Operational Semantics for Fixed-Point Logics on Constraint Databases , 2001, LPAR.

[13]  Jan Van den Bussche,et al.  Topological elementary equivalence of closed semi-algebraic sets in the real plane , 2000, Journal of Symbolic Logic.

[14]  Stéphane Grumbach,et al.  Constraint Databases , 1999, JFPLC.

[15]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[16]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[17]  David Gauld.  Differential topology: An introduction , 1982 .

[18]  Bart Kuijpers,et al.  Linear approximation of planar spatial databases using transitive-closure logic , 2000, PODS '00.

[19]  Jan Paredaens,et al.  Towards a theory of spatial database queries (extended abstract) , 1994, PODS.

[20]  Bart Kuijpers,et al.  On Expressing Topological Connectivity in Spatial Datalog , 1997, CDB.

[21]  A. Wilkie.  A theorem of the complement and some new o-minimal structures , 1999 .