Problems of Information Transmission

, Volume 45, Issue 1, pp 46–53 | Cite as

Perceptrons of large weight

  • V. V. Podolskii
Automata Theory


A threshold gate is a linear combination of input variables with integer coefficients (weights). It outputs 1 if the sum is positive. The maximum absolute value of the coefficients of a threshold gate is called its weight. A degree-d perceptron is a Boolean circuit of depth 2 with a threshold gate at the top and any Boolean elements of fan-in at most d at the bottom level. The weight of a perceptron is the weight of its threshold gate.

For any constant d ≥ 2 independent of the number of input variables n, we construct a degree-d perceptron that requires weights of at least \( n^{\Omega (n^d )} \); i.e., the weight of any degree-d perceptron that computes the same Boolean function must be at least \( n^{\Omega (n^d )} \). This bound is tight: any degree-d perceptron is equivalent to a degree-d perceptron of weight \( n^{O(n^d )} \). For the case of threshold gates (i.e., d = 1), the result was proved by Håstad in [2]; we use Håstad’s technique.


Serpentine Boolean Function Information Transmission Large Weight Order Number 
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Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsLomonosov Moscow State UniversityMoscowRussia

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