On the approximation of a random Boolean function by the set of quadratic forms

Abstract

We consider the problem of approximation of a random Boolean function by elements of the set of all Boolean functions of degree no greater than two, i.e., by the quadratic forms. It is proved that the Hamming distance from a random Boolean function of η variables to the set of all quadratic forms has in limit as η -» oo the double exponential distribution.

A local limit theorem for the distribution of a part of the spectrum of a random binary function

“…A local limit theorem for the distribution of some fixed number of coefficients was obtained by Ryazanov and Chechota in [6,7]. In the present paper, we prove a local limit theorem for the distribution of the vector F of an exponentially growing dimension M(n, k).…”

“…Combining Theorem 1 and formulas (10), (11), we can derive an asymptotic formula for the probabilities P(F = a) for integer-valued vectors of length M(n,k] satisfying conditions (12). In Ryazanov's paper [6], the following local limit theorem for the distribution of a part of the spectrum was proved (and generalized in [7]). Accordingly to formula (17) in [6], for any fixed r > 1 and n -> °°e xp(-Σ /-ι 0//2"-1 ) + 0(2-«/ 2 ) P{F/.…”

Limit Distribution of the Hamming Distance from the Random Boolean Function to the Set of Affine Functions

“…Therefore, Now the validity of (14) follows from Corollary 1 of Theorem 3 in [2] on the asymptotics of the probability for a sum of independent random m-dimensional vectors of a special form to get into the domain obtained by shifting the "positive octant" in R m by the vector whose components tend to +∞.…”

Bounds for the number of Boolean functions admitting affine approximations of a given accuracy

“….f; l i j / rgtj; (5) and the values of the partial sums over k from 1 to m D 1; 2; : : : ; 2 nC1 yield the upper (for odd m) and lower (for even m) bounds for jF V n 2 .r/j. So, in order to prove the theorem it suffices to obtain formulas for the terms in the right-hand side of (5) corresponding to k D 1; 2; 3 and show that for r < 2 n 2 C2 n 4 there exist no Boolean functions f in the r-neighbourhood of four (hence, of any number k > 4) different Boolean functions.…”

“…Thus, all terms of the inner sum in (5) corresponding to the case k D 2 coincide with jff 2 F V n 2 W .f; 0/ r; .f; l 2 / rgj; whereas the terms of the inner sum in (5) corresponding to the case k D 3 coincide with jff 2 F V n 2 W .f; 0/ r; .f; l 2 / r; .f; l 4 / rgj: The number of nonzero terms of the inner sum in (5) with k D 2 is equal to the number of pairs of different affine functions whose sum differs from the function l 1 .x/ 1, that is, to 4 2 n 2 (to each pair of different linear functions there corresponds four pairs of affine functions).…”