A Theorem on the Limiting Distribution for the Number of False Solutions of a System of Nonlinear Random Boolean Equations

Abstract

We prove that the distribution of the number of false solutions of a consistent system of nonlinear random Boolean equations with stochastically independent coefficients is asymptotically Poisson with parameter $2^m$ as the number n of unknowns tends to infinity. Our principal assumptions are: the distributions of the coefficients vary in a vicinity of the point $\oot\!;\,$ n and the number N of equations of the system differ by a constant m as $n\to\iy$; the system has a solution which contains $\rho(n)$ units, where $\rho(n)\ty$ as $n\to\iy$.