A better upper bound on weights of exact threshold functions

Abstract

A Boolean function is called an exact threshold function if it decides whether the input vector χ ∈ {0,1}n is on a hyperplane wT χ = t (w ∈ ℤn, t ∈ ℤ). In this paper we study the upper bound of elements in w required to represent any exact threshold function. Let k be the dimension of the linear subspace spanned by Boolean points on wT χ = t. We first give an upper bound O(n k) for constant k, which matches the lower bound in [2]. Then we prove an upper bound O(ko(k2)nk) for general cases, improving the result min{n2k, nn/2+1 in [2]. © 2011 Springer-Verlag.