The Complexity of SPP Formula Minimization

  • David Buchfuhrer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)


Circuit minimization is a useful procedure in the field of logic synthesis. Recently, it was proven that the minimization of ( ∨ , ∧ ,¬) formulae is hard for the second level of the polynomial hierarchy [BU08]. The complexity of minimizing more specialized formula models was left open, however. One model used in logic synthesis is a three-level model in which the third level is composed of parity gates, called SPPs. SPPs allow for small representations of Boolean functions and have efficient heuristics for minimization. However, little was known about the complexity of SPP minimization. Here, we show that SPP minimization is complete for the second level of the Polynomial Hierarchy under Turing reductions.


Boolean Function Full Version Negative Instance Logic Synthesis Prime Implicants 
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  1. [BCDV08]
    Bernasconi, A., Ciriani, V., Drechsler, R., Villa, T.: Logic minimization and testability of 2-SPP networks. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (to appear, 2008)Google Scholar
  2. [Bry86]
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Computers 35(8), 677–691 (1986)CrossRefMATHGoogle Scholar
  3. [BU08]
    Buchfuhrer, D., Umans, C.: The complexity of Boolean formula minimization. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 24–35. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. [CB02]
    Ciriani, V., Bernasconi, A.: 2-SPP: a practical trade-off between SP and SPP synthesis. In: 5th International Workshop on Boolean Problems (IWSBP 2002), pp. 133–140 (2002)Google Scholar
  5. [Cir03]
    Ciriani, V.: Synthesis of SPP three-level logic networks using affine spaces. IEEE Trans. on CAD of Integrated Circuits and Systems 22(10), 1310–1323 (2003)CrossRefGoogle Scholar
  6. [GHM08]
    Goldsmith, J., Hagen, M., Mundhenk, M.: Complexity of DNF minimization and isomorphism testing for monotone formulas. Information and Computation 206(6), 760–775 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. [JSA97]
    Jacob, J., Sivakumar, P.S., Agrawal, V.D.: Adder and comparator synthesis with exclusive-or transform of inputs. In: VLSI Design, pp. 514–515. IEEE Computer Society, Los Alamitos (1997)Google Scholar
  8. [LP99]
    Luccio, F., Pagli, L.: On a new Boolean function with applications. IEEE Trans. Computers 48(3), 296–310 (1999)MathSciNetCrossRefGoogle Scholar
  9. [Uma98]
    Umans, C.: The minimum equivalent DNF problem and shortest implicants. In: FOCS, pp. 556–563 (1998)Google Scholar
  10. [Uma99]
    Umans, C.: Hardness of approximating \(\Sigma_{2}^{p}\) minimization problems. In: FOCS, pp. 465–474 (1999)Google Scholar
  11. [Uma01]
    Umans, C.: The minimum equivalent DNF problem and shortest implicants. J. Comput. Syst. Sci. 63(4), 597–611 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. [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
  1. 1.Computer Science DepartmentCalifornia Institute of TechnologyPasadena

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