Optimal bounds for the approximation of boolean functions and some applications

This paper presents an optimal bound on the Shannon function L( n, m, ε) that gives the worstcase circuit-size complexity to approximate, within an approximation degree at least ε, partial boolean functions having n inputs and domain size m. That is L(n,m,ε) = Θ( mε 2 log(2 + mε 2) ) + O(n) . Our bound applies to any partial boolean function and any approximation degree, and thus completes the study of boolean function approximation introduced by Pippenger (1977). Our results give an upper bound for the hardness function h(ƒ), introduced by Nisan and Wigderson (1994), which denotes the minimum value l for which there exists a circuit of size at most l that approximates a boolean function ƒ with degree at least 1 l . Indeed, if H( n) denotes the maximum hardness value achieved by boolean functions with n inputs, we prove that for almost every n H(n)