Complexity of deciding Tarski algebra
Journal of Symbolic Computation
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
Testing stability by quantifier elimination
Journal of Symbolic Computation - Special issue: applications of quantifier elimination
On projection in CAD-based quantifier elimination with equational constraint
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Improved projection for cylindrical algebraic decomposition
Journal of Symbolic Computation
Computational Geometry Problems in REDLOG
Selected Papers from the International Workshop on Automated Deduction in Geometry
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Verification and synthesis using real quantifier elimination
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Variant quantifier elimination
Journal of Symbolic Computation
Critical points and Gröbner bases: the unmixed case
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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We study a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the ouput to be almost equivalent to the input. In a sense, we are strengthening the pre-condition and weakening the post-condition of the standard QE problem. The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the ouput formula is almost equivalent to the input formula. Thus, we propose to solve a variant of the initial quantifier elimination problem. We present an algorithm (VQE), that exploits the strengthened pre-condition and the weakened post-condition. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals. We find that the algorithm VQE can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack's scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However the current implementation of VQE can solve it in about 1 day.