Cylindrical Algebraic Sub-Decompositions

Cylindrical Algebraic Decompositions (CADs) have been studied since their creation as a tool for working with semi-algebraic sets and eliminating quantifiers of the reals. In this paper we are concerned with Cylindrical Algebraic Sub-Decompositions (sub-CADs), defined as subsets of CADs sufficient to describe the solutions for given formulae. We discuss several ways to construct sub-CADs, both from new work and the literature, and we demonstrate their interaction. We define a manifold sub-CAD as those cells in a CAD lying on a designated manifold and a layered sub-CAD as those cells with prescribed dimensions. We present algorithms to produce both and then describe how the ideas may be combined with each other, as well as their interaction with the recent theory of Truth-Table Invariant CAD (TTICAD). We give a complexity analysis for manifold sub-CADs and 1-layered manifold sub-CADs and present examples where these techniques can be of use, offering substantial savings on previous approaches. All concepts have been fully implemented in a Maple package and examples demonstrate that they can offer substantial savings.