Complex polynomials and circuit lower bounds for modular counting

Abstract

In this paper we study the power of constant-depth circuits containing negation gates, unobunded fan-in AND and OR gates, and a small number of MAJORITY gates. It is easy to show that a depth 2 circuit of size O(n) (where n is the number of inputs) containing O(n) MAJORITY gates can determine whether the sum of the input bits is divisible by k, for any fixed k>1, whereas it is known that this requires exponentialsize circuits if we have no MAJORITY gates. Our main result is that a constant-depth circuit of size $2^{n^{ \circ (1)} }$ containing n ^o(1) MAJORITY gates cannot determine if the sum of the input bits is divisible by k. This result was previously known only for k=2. To prove it for general k, we view a circuit as computing a function from {1,ω,..., ω^k−1}^m into the complex numbers. We are then able to approximately represent the behavior of the circuit by a multivariate complex polynomial of low degree. We then prove the main theorem by showing that the function taking (x ₁,..., x _m) ∈ {1,ω,..., ω^k−1}^m to x ₁... x _m does not admit the required sort of polynomial approximation.

Research supported by NSF Grant CCR-8714714.

Research supported by NSF Grant CCR-8902369.

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