Perceptrons of Large Weight

  • Vladimir V. Podolskii
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4649)


A threshold gate is a sum of input variables with integer coefficients (weights). It outputs 1 if the sum is positive. The maximal absolute value of coefficients of a threshold gate is called its weight. A perceptron of order d is a circuit of depth 2 having a threshold gate on the top level and any Boolean gates of fan-in at most d on the remaining level.

For every constant d ≥ 2 independent of the number of inputs n we exhibit a perceptron of order d that requires weights at least \(n^{\Omega(n^d)}\), that is, the weight of any perceptron of order d computing the same Boolean function is at least \(n^{\Omega(n^d)}\). This bound is tight: every perceptron of order d is equivalent to a perceptron of order d and weight \(n^{O(n^d)}\). In the case of threshold gates (i.e. d = 1) the result was established by Håstad in [1]; we use Håstad’s techniques.


Boolean Function Ordinal Number Bottom Level Polynomial Hierarchy Threshold Gate 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Vladimir V. Podolskii
    • 1
  1. 1.Moscow State University 

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