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 x ∈ {0,1}ⁿ is on a hyperplane w ^T x= 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 w ^T x= 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 $O(k^{O(k^2)}n^k)$ for general cases, improving the result $\min\{n^{2^k},n^{n/2+1}\}$ in [2].

This work was supported in part by the National Natural Science Foundation of China Grant 60603005, 61033001, 61061130540, the National Basic Research Program of China Grant 2007CB807900, 2007CB807901, and Tsinghua University Initiative Scientific Research Program 2009THZ02120.