On K-wise Independent Distributions and Boolean Functions

TL;DR

This paper investigates how Boolean functions behave under k-wise independent input distributions, identifying the minimum k needed for such distributions to mimic full independence, using novel analytical tools.

Contribution

It provides a systematic analysis of k-wise independence effects on Boolean functions, introducing new techniques from the classical moment problem to this area.

Findings

Certain functions are fooled by surprisingly low k-wise independence

The analysis reveals unexpected properties of well-known Boolean functions

New connections between moment problems and Boolean function analysis

Abstract

We pursue a systematic study of the following problem. Let f:{0,1}^n -> {0,1} be a (usually monotone) Boolean function whose behaviour is well understood when the input bits are identically independently distributed. What can be said about the behaviour of the function when the input bits are not completely independent, but only k-wise independent, i.e. every subset of k bits is independent? more precisely, how high should k be so that any k-wise independent distribution "fools" the function, i.e. causes it to behave nearly the same as when the bits are completely independent? We analyze several well known Boolean functions (including AND, Majority, Tribes and Percolation among others), some of which turn out to have surprising properties. In some of our results we use tools from the theory of the classical moment problem, seemingly for the first time in this subject, to shed light on these questions.