On the size of probabilistic formulae

Session 5B
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)


We investigate size complexity for different models of probabilistic formulae. Relating formula-size to one-way communication complexity we devise a lower bound method for probabilistic formulae based on the VC-dimension and the Nečiporuk lower bound and give for the first time lower bounds on the size of 2-sided bounded error, Monte Carlo, and Las Vegas formulae over an arbitrary basis. We show that for the Boolean matrix product Monte Carlo probabilistic formulae are smaller by a factor of Ω/trn than Las Vegas formulae and prove an analogous gap between two sided bounded error and Monte Carlo formulae. This is the maximal gap between probabilistic and deterministic formulae provable using the Nečiporuk method. We also consider a function for which two-sided-error probabilism does not help. Furthermore we investigate the effect of restricted access to randomness on formula size, and study generalizations of matrix products which may be candidates for proving larger gaps between probabilism and determinism.


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  1. [BNS92]
    L. Babai, N. Nisan, M. Szegedy. Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-offs. Journ. of Computer and System Sciences, vol.45, pp. 204–232, 1992.Google Scholar
  2. [B85]
    R.B. Boppana. Amplification of probabilistic Boolean formulas. 26th Symp. Found. Comput. Science, pp. 20–29, 1985.Google Scholar
  3. [BS90]
    R.B. Boppana, M. Sipser. The Complexity of Finite Functions. Handbook of Theoretical Computer Science, vol. A. Elsevier, 1990.Google Scholar
  4. [DZ97]
    M. Dubiner, U. Zwick. Amplification by Read-Once Formulae. SIAM Journal Comput., vol.26, pp. 15–38, 1997.Google Scholar
  5. [DHRS97]
    P. Duriš, J. Hromkovič, J.D.P. Rolim, G. Schnitger. Las Vegas Versus Determinism for One-way Communication Complexity, Finite Automata, and Polynomial-time Comp. 14th Symp. on Theor. Aspects of Comp. Science, pp. 117–128, 1997.Google Scholar
  6. [H97]
    J. Hromkovîc. Communication Complexity and Parallel Computing. Springer, 1997.Google Scholar
  7. [KNR95]
    I. Kremer, N. Nisan, D. Ron. On Randomized One-Round Communication Complexity. 27th Symp. Theory of Comput., pp. 596–605, 1995.Google Scholar
  8. [KN96]
    E. Kushilevitz, N. Nisan. Communication Complexity. Cambridge University Press, 1996.Google Scholar
  9. [KW90]
    M. Karchmer, A. Wigderson. Monotone Circuits for Connectivity Require Super-Log. Depth. SIAM Journ. Discrete Math., vol.3, pp. 718–727, 1990.Google Scholar
  10. [N66]
    E.I. Nečiporuk. A Boolean function. Sov.Math.Dokl., vol.7, pp. 999–1000, 1966.Google Scholar
  11. [Ne91]
    I. Newman. Private vs. Common Random Bits in Communication Complexity. Information Processing Letters, vol.39, pp.67–71, 1991.Google Scholar
  12. [V84]
    L.G. Valiant. Short monotone formulae for the majority function. Journal of Algorithms, vol.5, pp. 363–366, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  1. 1.Fachbereich InformatikJohann-Wolfgang-Goethe-Universität FrankfurtFrankfurt am MainGermany

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