Finding Efficient Circuits Using SAT-Solvers

  • Arist Kojevnikov
  • Alexander S. Kulikov
  • Grigory Yaroslavtsev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5584)


In this paper we report preliminary results of experiments with finding efficient circuits (over binary bases) using SAT-solvers. We present upper bounds for functions with constant number of inputs as well as general upper bounds that were found automatically. We focus mainly on MOD-functions. Besides theoretical interest, these functions are also interesting from a practical point of view as they are the core of the residue number system. In particular, we present a circuit of size 3n + c over the full binary basis computing \({\rm MOD}_3^n\).


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  1. 1.
    Chin, A.: On the depth complexity of the counting functions. Information Processing Letters 35, 325–328 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Eén, N.: Practical SAT — a tutorial on applied satisfiability solving. Slides of invited talk at FMCAD (2007)Google Scholar
  3. 3.
    Estrada, G.G.: A note on designing logical circuits using SAT. In: Tyrrell, A.M., Haddow, P.C., Torresen, J. (eds.) ICES 2003. LNCS, vol. 2606, pp. 410–421. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Fischer, M.J., Meyer, A.R., Paterson, M.S.: Ω(n logn) lower bounds on length of Boolean formulas. SIAM Journal on Computing 11, 416–427 (1982)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fogel, D.B.: Evolutionary computation: The fossil record. IEEE Press, New York (1998)CrossRefMATHGoogle Scholar
  6. 6.
    Kamath, A.P., Karmarkar, N.K., Ramakrishnan, K.G., Resende, M.G.C.: An interior point approach to boolean vector function synthesis. In: Proceedings of the 36th International Midwest Symposium on Circuits and Systems (MSCAS 1993), pp. 185–189 (1993)Google Scholar
  7. 7.
    Khatri, S., Shenoy, N.: Logic synthesis. In: Scheffer, L., Lavagno, L., Martin, G. (eds.) Electronic Design Automation For Integrated Circuits Handbook. CRC Press, Taylor & Francis Group (2006)Google Scholar
  8. 8.
    Khrapchenko, V.M.: Complexity of the realization of a linear function in the case of Π-circuits. Math. Notes Acad. Sciences 9, 21–23 (1971)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Koren, I.: Computer Arithmetic Algorithms. Prentice Hall, Englewood Cliffs (1993)MATHGoogle Scholar
  10. 10.
    Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Systems Technical Journal 38, 985–999 (1959)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Massey, J.L.: The difficulty with difficulty. In: A Guide to the Transparencies from the EUROCRYPT 1996 IACR Distinguished Lecture (1996)Google Scholar
  12. 12.
    McCluskey, E.J.: Logic Design Principles: with emphasis on testable semicustom circuits. Prentice-Hall, Englewood Cliffs (1986)Google Scholar
  13. 13.
    Nigmatullin, R.G.: Slognost’ bulevikh funktsii. Moskva, Nauka (1991) (in Russian)Google Scholar
  14. 14.
    Paterson, M.S., Zwick, U.: Shallow circuits and concise formulae for multiple addition and multiplication. Computational Complexity 3, 262–291 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Prasad, M.R., Biere, A., Aarti, G.: A survey of recent advances in SAT-based formal verification. International Journal on Software Tools for Technology Transfer 7(2), 156–173 (2005)CrossRefGoogle Scholar
  16. 16.
    Razborov, A.A.: Lower bounds for the monotone complexity of some Boolean functions. Soviet Math. Doklady 31, 354–357 (1985)MATHGoogle Scholar
  17. 17.
    Gelatt, C.D., Kirkpatrick, S., Vecchi, M.P.: Optimization by simulated annealing. Science, New Series 220(4598), 671–680 (1983)MathSciNetMATHGoogle Scholar
  18. 18.
    Schnorr, C.: Zwei lineare untere Schranken für die Komplexität Boolescher Funktionen. Computing 13, 155–171 (1974)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Stockmeyer, L.J.: On the combinational complexity of certain symmetric Boolean functions. Mathematical Systems Theory 10, 323–336 (1977)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    van Leijenhorst, D.C.: A note on the formula size of the “mod k” functions. Information Processing Letters 24, 223–224 (1987)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Williams, R.: Applying practice to theory. ACM SIGACT News 39(4), 37–52 (2008)CrossRefGoogle Scholar
  22. 22.
    Zwick, U.: A 4n lower bound on the combinational complexity of certain symmetric boolean functions over the basis of unate dyadic Boolean functions. SIAM Journal on Computing 20, 499–505 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Arist Kojevnikov
    • 1
  • Alexander S. Kulikov
    • 2
  • Grigory Yaroslavtsev
    • 3
  1. 1.OneSpin Solutions GmbHGermany
  2. 2.St. Petersburg Department of Steklov Institute of MathematicsRussia
  3. 3.Academic Physics and Technology University of the RASRussia

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