Fast simplifications for Tarski formulas
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Black-box/white-box simplification and applications to quantifier elimination
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Fast simplifications for Tarski formulas based on monomial inequalities
Journal of Symbolic Computation
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The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be $\Sigma_2^P$-complete and indeed appears as an open problem in Garey and Johnson [GJ79]. The depth-2 variant was only shown to be $\Sigma_2^P$-complete in 1998 [Uma98], and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is $\Sigma_2^P$-complete under Turing reductions for all k 驴 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is $\Sigma_2^P$-complete under Turing reductions.