The complexity of Boolean formula minimization

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 -complete and indeed appears as an open problem in Garey and Johnson (1979) [5], . The depth-2 variant was only shown to be -complete in 1998 (Umans (1998) [13], , Umans (2001) [15]) 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 -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 -complete under Turing reductions.