Optimal bounds on the approximation of boolean functions with consequences on the concept of hardness

Abstract

We prove an optimal bound for the function L(n, m, e) that gives the worst-case circuit-size complexity to approximate partial boolean functions having n inputs and domain size m within degree at least e. Our bound applies to any partial boolean function and any approximation degree, completing the study of boolean function approximation introduced in [15]. We also provide the approximation degree (i.e. the value e) achieved by polynomial size circuits on a ‘random’ boolean function. Our results give a new upper bound for the hardness function h(f), the function denoting the minimum value l for which there exists a circuit of size at most l that approximates a boolean function f with degree at least 1/l [14]. The contribution in the proof of the upper bound for L(n, m, e) can be viewed as a set of technical results that globally show how boolean linear operators are “well” distributed over the class of 4-regular domains. We show how to apply this property to approximate partial boolean functions on general domains.