Weights of Exact Threshold Functions

Abstract

We consider Boolean exact threshold functions defined by linear equations, and in general degree d polynomials. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close. In the linear case in particular they are almost matching. This quantity is the same as the maximum magnitude of integer coefficients of linear equations required to express every possible intersection of a hyperplane in R n and the Boolean cube {0, 1} n , or in the general case intersections of hypersurfaces of degree d in R n and the Boolean cube {0, 1} n . In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension k of the affine subspace spanned by the solutions, and give upper and lower bounds in this case as well. Our bounds here in terms of k leave a substantial gap, a challenge for future work.

Lower Bound on Weights of Large Degree Threshold Functions

“…For this result another proof was given in [1]. Some other results on large degree threshold gates which are not directly connected to the problem we consider have appeared in [5,2,19,14,16,4].…”

Patience of matrix games

“…A full sign pattern is a matrix with entries from {−1, 1}. A pair of vectors σ (1) , σ (2) with entries from {−1, 1} gives rise to a full sign pattern σ (1) (σ (2) ) T We shall call a full sign pattern of this form a block checkerboard sign pattern.…”

“…The problem of constructing (0, 1) or (−1, 1) matrices A for which χ(A) is large was considered first by Graham and Sloane [8], and later by Alon and Vũ [1]. Such matrices have besides the direct application of constructing ill-conditioned matrices, several applications such as flat simplices, coin weighing, indecomposable hypergraphs, and weights of Boolean threshold functions [8,12,1,17,2].…”

“…For m ≥ 6, we in fact prove that the first block of columns is of size 2m − 1. Thus Σ = σ (1) (σ (2) ) T , where σ (1) i = (−1) |αi| and σ (2) = ( 2m−1 −1, . .…”

Bounds for the Diameter of the Weight Polytope

“…Bounds for the necessary magnitude of integer weighted are studied in the literature, see e.g. [1] and the references therein.…”