Bent functions and random Boolean formulas

Let alpha be a nonlinear boolean connective with equal number of zero and unit entries in its table. We study an iterative process of combining randomly chosen boolean functions from some starting set G via the connective alpha. We suppose that all functions in G are defined on the same finite nonempty domain M. We are interested in the situations, when the process converges to the uniform distribution on {0, 1}(M). We classify the boolean connectives alpha according to the asymptotic rate of this convergence. Although the probability of any function in our process converges to (1/2)(\M\), there are some differences in the terms of lower order of magnitude. If M is the boolean cube of an even dimension n and G is the set of all linear boolean functions of n variables and the connective alpha belongs to the class of the lowest possible rate of convergence in the above-mentioned classification, we can express the main nonconstant term of the asymptotic expansion of the probability of occurring of a single function f through the Fourier transform of f. Using this, we prove that the bent functions achieve asymptotically the minimal probability of occurring among all boolean functions. At the same time, the linear functions achieve asymptotically the maximal probability.