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 \(\Sigma_2^P\)-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.
KeywordsPositive Instance Boolean Formula Negative Instance Boolean Circuit Equivalent Formula
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- [DGK94]Devadas, S., Ghosh, A., Keutzer, K.: Logic synthesis. McGraw-Hill, New York (1994)Google Scholar
- [HW97]Hemaspaandra, E., Wechsung, G.: The minimization problem for Boolean formulas. In: FOCS, pp. 575–584 (1997)Google Scholar
- [KC00]Kabanets, V., Cai, J.-Y.: Circuit minimization problem. In: STOC, pp. 73–79 (2000)Google Scholar
- [MS72]Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: FOCS, pp. 125–129. IEEE, Los Alamitos (1972)Google Scholar
- [Uma98]Umans, C.: The minimum equivalent DNF problem and shortest implicants. In: FOCS, pp. 556–563 (1998)Google Scholar
- [Uma99a]Umans, C.: Hardness of approximating \(\Sigma_2^P\) minimization problems. In: FOCS, pp. 465–474 (1999)Google Scholar