On the minimization of SOPs for bi-decomposable functions

A function f is AND bi-decomposable if it can be written as f(X ₁,X₂)=h₁(X₁)h₂(X ₂). In this case, a sum-of-products expression (SOP) for f is obtained from minimum SOPs (MSOP) for h₁ and h₂ by applying the law of distributivity. If the result is an MSOP, then the complexity of minimization is reduced. However, the application of the law of distributivity to MSOPs for h₁ and h₂ does not always produce an MSOP for f. We show an incompletely specified function of n(n-1) variables that requires at most n products in an MSOP, while 2^n-1 products are required by minimizing the component functions separately. We introduce a new class of logic functions, called orthodox functions, where the application of the law of distributivity to MSOPs for component functions of f always produces an MSOP for f. We show that orthodox functions include all functions with three of fewer variables, all symmetric functions, all unate functions, many benchmark functions, and few random functions with many variables