Simulating Threshold Circuits by Majority Circuits

We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size depth 2 majority circuit. In general we show that a polynomial-size, depth- d threshold circuit can be simulated uniformly by a polynomial-size majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10] that a non-uniform simulation exists. Our construction answers two open questions posed in [10]: we give an explicit construction whereas [10] uses a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (the simulation in [10] gives polynomial-size circuits only when d is constant)