A non-probabilistic switching lemma for the Sipser function

  • Sorin Istrail
  • Dejan Zivkovic
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)


Valiant [12] showed that the clique function is structurally different than the majority function by establishing the following “switching lemma”: Any function f whose set of prime implicants is a large enough subset of the set of cliques (and thus requiring big Σ2-circuits), has a large set of prime clauses (i.e., big Π2-circuits). As a corollary, an exponential lower bound was obtained for monotone ΣΠΣ-circuits computing the clique function. The proof technique is the only non-probabilistic super polynomial lower bound method from the literature. We prove, by a non-probabilistic argument as well, a similar switching lemma for the NC1-complete Sipser function. Using this we then show that a monotone depth-3 (i.e., ΣΠΣ or ΠΣΠ) circuit computing the Sipser function must have super quasipolynomial size. Moreover, any depth-d quasipolynomial size non-monotone circuit computing the Sipser function has a depth-(d—1) gate computing a function with exponentially many both prime implicants and (monotone) prime clauses. These results are obtained by a top-down analysis of the circuits.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E. Allender, “A note on the power of threshold circuits”, Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pp. 580–584, 1989.Google Scholar
  2. [2]
    D. A. Barrington, “Bounded-width polynomial-size branching programs recognize exactly those languages in NC1”, Journal of Computer and System Sciences, Vol. 38, pp. 150–164, 1989.Google Scholar
  3. [3]
    R. Beigel and J. Tarui, “On ACC”, Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pp. 783–792, 1991.Google Scholar
  4. [4]
    R. B. Boppana and M. Sipser, “The Complexity of Finite Functions”, Handbook of Theoretical Computer Science, Vol. A (J. van Leeuwen, ed., North-Holland, Amsterdam), pp. 757–804, 1990.Google Scholar
  5. [5]
    J. Hastad, “Almost optimal lower bounds for small-depth circuits”, Proceedings of the 18th ACM Symposium on Theory of Computing, pp. 6–20, 1986.Google Scholar
  6. [6]
    S. Istrail and D. Zivkovic, “A non-probabilistic switching lemma for the Sipser function”, Wesleyan University, CS/TR-92-1, 1992.Google Scholar
  7. [7]
    M. Karchmer and A. Wigderson, “Monotone circuits for connectivity require super-logarithmic depth”, Proceedings of the 20th ACM Symposium on Theory of Computing, pp. 539–550, 1988.Google Scholar
  8. [8]
    D. Mundici, “Functions computed by monotone Boolean formulas with no repeated variables”, Theoretical Computer Science, Vol. 66, pp. 113–114, 1989.Google Scholar
  9. [9]
    A. A. Razborov, “Lower bounds on the monotone complexity of some Boolean functions”, Doklady Akademii Nauk SSSR, Vol. 281(4), pp. 798–801, 1985 (in Russian). English translation in Soviet Mathematics Doklady, Vol. 31, pp. 354–357, 1985.Google Scholar
  10. [10]
    A. A. Razborov, “Lower bounds on the size of bounded depth networks over a complete basis with logical addition”, Matematicheskie Zametki, Vol. 41(4), pp. 598–607, 1987 (in Russian). English translation in Mathematical Notes of the Academy of Sciences of the USSR, Vol. 41(4), pp. 333–338, 1987.Google Scholar
  11. [11]
    S. Skyum and L. G. Valiant, “A complexity theory based on Boolean algebra”, Journal of the ACM, Vol. 22, pp. 484–504, 1985.Google Scholar
  12. [12]
    L. G. Valiant, “Exponential lower bounds for restricted monotone circuits”, Proceedings of the 15th ACM Symposium on Theory of Computing, pp. 110–117, 1983.Google Scholar
  13. [13]
    I. Wegener, The Complexity of Boolean Functions, Wiley-Teubner, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Sorin Istrail
    • 1
  • Dejan Zivkovic
    • 1
  1. 1.Department of MathematicsWesleyan UniversityMiddletownUSA

Personalised recommendations