Improved Projection for Cylindrical Algebraic Decomposition

Abstract

McCallum’s projection operator for cylindrical algebraic decomposition (CAD) represented a huge step forward for the practical utility of the CAD algorithm. This paper presents a simple theorem showing that the mathematics in McCallum’s paper actually point to a better projection operator than he proposes—a reduced McCallum projection. The reduced projection has the potential to not simply speed up CAD computation for problems that are currently solvable in practice, but actually increase the scope of problems that can realistically be attacked via CADs. Additionally, the same methods are used to show that McCallum’s projection can be reduced still further when CAD is applied to certain types of commonly occurring quantifier elimination problems.