On the correlation between Boolean functions of sequences of random variables

In this paper, we establish a new inequality tying together the effective length and the maximum correlation between the outputs of an arbitrary pair of Boolean functions which operate on two sequences of correlated random variables. We derive a new upper-bound on the correlation between the outputs of these functions. The upper-bound is useful in various disciplines which deal with common-information. We build upon Witsenhausen's [2] bound on maximum-correlation. The previous upper-bound did not take the effective length of the Boolean functions into account. One possible application of the new bound is to characterize the communication-cooperation tradeoff in multi-terminal communications. In this problem, there are lower-bounds on the effective length of the Boolean functions due to the rate-distortion constraints in the problem, as well as lower bounds on the output correlation at different nodes due to the multi-terminal nature of the problem.