The Complexity of Boolean Formula Minimization

Abstract

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 ≥ 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.