Practical implementation of block-structured quantifier elimination algorithms

Develop practical implementations of quantifier elimination algorithms over the reals that exploit a block structure and achieve the theoretical complexity guarantees of being doubly exponential in the number of quantifier alternations and exponential in the total number of unknowns, thereby realizing these algorithms in practice.

Background

The paper reviews advances in quantifier elimination (QE) that leverage block structures of quantified variables to improve complexity, tracing developments from Grigoriev to Basu–Pollack–Roy. These methods yield algorithms with complexity doubly exponential in the number of quantifier alternations and exponential in the total number of unknowns.

Despite these theoretical developments, the authors explicitly note that making such QE algorithms practical remains unresolved. Their work focuses instead on leveraging the specific structure of parametric linear matrix inequalities to obtain implementable procedures under genericity assumptions, leaving the general practical implementation of the aforementioned QE algorithms as an open problem.