The Complexity of Boolean Formula Minimization

  • David Buchfuhrer
  • Christopher Umans
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [DGK94]
    Devadas, S., Ghosh, A., Keutzer, K.: Logic synthesis. McGraw-Hill, New York (1994)Google Scholar
  2. [GJ79]
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)MATHGoogle Scholar
  3. [HW97]
    Hemaspaandra, E., Wechsung, G.: The minimization problem for Boolean formulas. In: FOCS, pp. 575–584 (1997)Google Scholar
  4. [KC00]
    Kabanets, V., Cai, J.-Y.: Circuit minimization problem. In: STOC, pp. 73–79 (2000)Google Scholar
  5. [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
  6. [Sto76]
    Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)MathSciNetCrossRefMATHGoogle Scholar
  7. [Uma98]
    Umans, C.: The minimum equivalent DNF problem and shortest implicants. In: FOCS, pp. 556–563 (1998)Google Scholar
  8. [Uma99a]
    Umans, C.: Hardness of approximating \(\Sigma_2^P\) minimization problems. In: FOCS, pp. 465–474 (1999)Google Scholar
  9. [Uma99b]
    Umans, C.: On the Complexity and Inapproximability of Shortest Implicant Problems. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 687–696. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. [UVSV06]
    Umans, C., Villa, T., Sangiovanni-Vincentelli, A.L.: Complexity of two-level logic minimization. IEEE Trans. on CAD of Integrated Circuits and Systems 25(7), 1230–1246 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • David Buchfuhrer
    • 1
  • Christopher Umans
    • 1
  1. 1.Computer Science DepartmentCalifornia Institute of TechnologyPasadena

Personalised recommendations