Algebraic characteristics and satisfiability threshold of random Boolean equations

The satisfiability of a class of random Boolean equations named massive algebraic system septated to linear and nonlinear subproblems is studied in this paper. On one hand, the correlation between the magnetization of generators and the clustering of solutions of the linear subproblem is investigated by analyzing the Gaussian elimination process. On the other hand, the characteristics of maximal elements of solutions of the nonlinear subproblem are studied by introducing the partial order among solutions. Based on the algebraic characteristics of these two subproblems, the upper and lower bounds of satisfiability threshold of massive algebraic system are obtained by unit-clause propagation and leaf-removal process, and coincide as the ratio of nonlinear equations q > 0.739 in which analytical values of the satisfiability threshold can be derived. Furthermore, a complete algorithm with heuristic decimation is proposed to observe the approximation of the satisfiability threshold, which performs more efficiently than the classical ones.