A Poly-algorithmic Approach to Quantifier Elimination

Cylindrical Algebraic Decomposition (CAD) was the first practical means for doing Real Quantifier Elimination (QE), and is still a major method. Nevertheless, its complexity is inherently doubly exponential in the number of variables. Where applicable, virtual term substitution (VTS) is more effective, turning a QE problem in n variables to one in $n-1$ variables in one application, and so on. Hence there is scope for hybrid methods: doing VTS where possible then using CAD. This paper describes such a poly-algorithmic implementation, based on the second author’s Ph.D. thesis. The version of CAD used is based on a new implementation of Lazard’s method, with improvements to handle equational constraints, similar to the equational constraint handling in previous CAD algorithms.