Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
The complexity of linear problems in fields
Journal of Symbolic Computation
Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
An improvement of the projection operator in cylindrical algebraic decomposition
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Simplification of quantifier-free formulae over ordered fields
Journal of Symbolic Computation - Special issue: applications of quantifier elimination
Simple CAD construction and its applications
Journal of Symbolic Computation
Improved projection for cylindrical algebraic decomposition
Journal of Symbolic Computation
The complexity of quantifier elimination and cylindrical algebraic decomposition
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
The Complexity of Boolean Formula Minimization
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Fast simplifications for Tarski formulas
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Fast simplifications for Tarski formulas based on monomial inequalities
Journal of Symbolic Computation
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This paper describes a new method for simplifying Tarski formulas. The method combines simplifications based purely on the factor structure of inequalities ("black-box" simplification) with simplifications that require reasoning about the factors themselves. The goal is to produce a simplification procedure that is very fast, so that it can be applied --- perhaps many, many times --- within other algorithms that compute with Tarski formulas without ever slowing them down significantly, but which also produces useful simplification in a substantial number of cases. The method has been implemented and integrated into implementations of two important algorithms: quantifier elimination by virtual term substitution, and quantifier elimination by cylindrical algebraic decomposition. The paper reports on how the simplification method has been integrated with these two algorithms, and reports experimental results that demonstrate how their performance is improved.