A better upper bound on weights of exact threshold functions

Authors:
Xue Chen;Guangda Hu;Xiaoming Sun
Affiliations:
Department of Computer Science and Technology, Tsinghua University;Department of Computer Science and Technology, Tsinghua University;Institute for Theoretical Computer Science and Center for Advanced Study, Tsinghua University
Venue:
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Year:
2011

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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 wT 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 wTx = t. We first give an upper bound O(nk) 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].