  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.
