The Development of Suitable Inequalities and Their Application to Systems of Logical Equations

In this paper, two not-difficult inequalities are invented and proved in detail, which adequately describe the behavior of discrete logical functions xor(x1, x2,…, xn) and and(x1, x2,…, xn). Based on these proven inequalities, infinitely differentiable extensions of the logical functions xor(x1, x2,…, xn) and and(x1, x2,…, xn) were defined for the entire ℝn. These suitable extensions were applied to systems of logical equations. Specifically, the system of m logical equations in a constructive way without adding any equations (not field equations and no others) is transformed in ℝn first into an equivalent system of m smooth rational equations (SmSRE) so that the solution of SmSRE can be reduced to the problem minimization of the objective function, and any numerical optimization methods can be applied since the objective function will be infinitely differentiable. Again, we transformed SmSRE into an equivalent system of m polynomial equations (SmPE). This means that any symbolic methods for solving polynomial systems can be used to solve and analyze an equivalent SmPE. The equivalence of these systems has been proved in detail. Based on these proofs and results, in the next paper, we plan to study the practical applicability of numerical optimization methods for SmSRE and symbolic methods for SmPE.