Model-based construction of Open Non-uniform Cylindrical Algebraic Decompositions

In this paper we introduce the notion of an Open Non-uniform Cylindrical Algebraic Decomposition (NuCAD), and present an efficient model-based algorithm for constructing an Open NuCAD from an input formula. A NuCAD is a generalization of Cylindrical Algebraic Decomposition (CAD) as defined by Collins in his seminal work from the early 1970s, and as extended in concepts like Hong’s partial CAD. A NuCAD, like a CAD, is a decomposition of R into cylindrical cells. But unlike a CAD, the cells in a NuCAD need not be arranged cylindrically. It is in this sense that NuCADs are not uniformly cylindrical. However, NuCADs — like CADs — carry a tree-like structure that relates different cells. It is a very different tree but, as with the CAD tree structure, it allows some operations to be performed efficiently, for example locating the containing cell for an arbitrary input point.