Explicit Constructions of Depth-2 Majority Circuits for Comparison and Addition

All Boolean variables here range over the two element set {−1, 1}. Given n Boolean variables x1, . . . , xn, a non-monotone MAJORITY gate (in the variables xi) is a Boolean function whose value is the sign of ∑n i=1 ixi, where each i is either 1 or −1. The COMPARISON function is the Boolean function of two n-bits integers X and Y whose value is −1 iff X ≥ Y . We construct an explicit sparse polynomial whose sign computes this function. Similar polynomials are constructed for computing all the bits of the summation of the two numbers X and Y . This supplies explicit constructions of depth-2 polynomial-size circuits computing these functions, which use only non-monotone MAJORITY gates. These constructions are optimal in terms of the depth and can be used to obtain the best known explicit constructions of MAJORITY circuits for other functions like the product of two n-bit numbers and the maximum of n nbit numbers. A crucial ingredient is the construction of a discrete version of a sparse “delta polynomial”—one that has a large absolute value for a single assignment and extremely small absolute values for all other assignments. ∗Research supported in part by the Fund for Basic Research administered by the Israel Academy of Sciences