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.


Positive Instance Boolean Formula Negative Instance Boolean Circuit Equivalent Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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