# 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, ε)* that gives the worst-case circuit-size complexity to approximate partial boolean functions having *n* inputs and domain size *m* within degree at least *ε*. 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 *ε*) 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, ε)* 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.

## Keywords

Linear Operator Boolean Function Domain Size General Domain Pseudorandom Generator## Preview

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