On the Incompressibility of Monotone DNFs

  • Matthias P. Krieger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3623)


We prove optimal lower bounds for multilinear circuits and for monotone circuits with bounded depth. These lower bounds state that, in order to compute certain functions, these circuits need exactly as many OR gates as the respective DNFs. The proofs exploit a property of the functions that is based solely on prime implicant structure. Due to this feature, the lower bounds proved also hold for approximations of the considered functions that are similar to slice functions. Known lower bound arguments cannot handle these kinds of approximations. In order to show limitations of our approach, we prove that cliques of size n–1 can be detected in a graph with n vertices by monotone formulae with O(log n) OR gates.

Our lower bound for multilinear circuits improves a lower bound due to Borodin, Razborov and Smolensky for nondeterministic read-once branching programs computing the clique function.


Boolean Function Boolean Circuit Prime Implicants Slice Function Monotone Circuit 
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. 1.
    Razborov, A.: Lower bounds for the monotone complexity of some Boolean functions. Sov. Math., Dokl. 31, 354–357 (1985)zbMATHGoogle Scholar
  2. 2.
    Karchmer, M., Wigderson, A.: Monotone circuits for connectivity require superlogarithmic depth. SIAM J. Discrete Math. 3, 255–265 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Razborov, A.: Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica 10, 81–93 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gál, A.: A characterization of span program size and improved lower bounds for monotone span programs. Comput. Complexity 10, 277–296 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gál, A., Pudlák, P.: A note on monotone complexity and the rank of matrices. Inf. Process. Lett. 87, 321–326 (2003)zbMATHCrossRefGoogle Scholar
  6. 6.
    Wegener, I.: The complexity of Boolean functions. Wiley-Teubner Series in Computer Science. John Wiley & Sons Ltd., Chichester (1987)zbMATHGoogle Scholar
  7. 7.
    Dunne, P.E.: The complexity of Boolean networks. APIC Studies in Data Processing, vol. 29. Academic Press Ltd., London (1988)zbMATHGoogle Scholar
  8. 8.
    Savage, J.E.: Models of computation: Exploring the power of computing. Addison- Wesley Publishing Company, Reading 1998)Google Scholar
  9. 9.
    Sengupta, R., Venkateswaran, H.: Multilinearity can be exponentially restrictive (preliminary version). Technical Report GIT-CC-94-40, Georgia Institute of Technology. College of Computing (1994)Google Scholar
  10. 10.
    Ponnuswami, A.K., Venkateswaran, H.: Monotone multilinear boolean circuits for bipartite perfect matching require exponential size. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 460–468. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Wegener, I.: Branching programs and binary decision diagrams. Theory and applications. SIAM Monographs on Discrete Mathematics and Applications (2000)Google Scholar
  12. 12.
    Nisan, N., Wigderson, A.: Lower bounds on arithmetic circuits via partial derivatives. Comput. Complexity 6, 217–234 (1996/1997)Google Scholar
  13. 13.
    Raz, R.: Multi-linear formulas for permanent and determinant are of superpolynomial size. In: Babai, L. (ed.) STOC, pp. 633–641. ACM, New York (2004)Google Scholar
  14. 14.
    Raz, R.: Multilinear-NC1 ≠ Multilinear-NC2. In: FOCS, pp. 344–351. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  15. 15.
    Borodin, A., Razborov, A.A., Smolensky, R.: On lower bounds for read-k-times branching programs. Comput. Complexity 3, 1–18 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Andreev, A.: On a method for obtaining lower bounds for the complexity of individual monotone functions. Sov. Math., Dokl. 31, 530–534 (1985)zbMATHGoogle Scholar
  17. 17.
    Grigni, M., Sipser, M.: Monotone complexity. In: Paterson, M.S. (ed.) Boolean function complexity. London Mathematical Society Lecture Note Series, vol. 169, pp. 57–75. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  18. 18.
    van Lint, J.H.: Introduction to coding theory. Graduate Texts in Mathematics, vol. 86. Springer, New York (1982)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Matthias P. Krieger
    • 1
  1. 1.Institut für Informatik, Lehrstuhl für Theoretische InformatikJohann Wolfgang Goethe-Universität FrankfurtFrankfurt am MainGermany

Personalised recommendations