Extend threshold circuit lower bounds to arbitrary weights and depth ≥ 3

Extend the exponential lower bound on the complexity of depth-2 threshold circuits with polynomially bounded weights, established by Hajnal, Maass, Pudlák, Szegedy, and Turán, to (i) depth-2 threshold circuits with arbitrary weights and (ii) threshold circuits with depth at least 3.

Background

In the review of pre-2010 theoretical results, the paper highlights a seminal contribution by Hajnal, Maass, Pudlák, Szegedy, and Turán that proved an exponential lower bound on the complexity of depth-2 threshold (küszöbfüggvény) circuits under the assumption of polynomially bounded weights.

The authors explicitly state that extending this lower bound remains unresolved in two directions: allowing arbitrary (unbounded) weights even at depth 2, and increasing circuit depth to at least 3. This situates the open problem within classical circuit complexity and the theoretical paper of neural (threshold) networks.