Generalized Lattices of Boolean Functions Utilized for Derivative Operations

This paper explores lattices of Boolean functions which are built by derivative operations of the Boolean Differential Calculus [2], [8], [9]. Such operations are needed to design circuits with short delay and low power consumption [11] as well as to calculate minimal complete sets of fitting test patterns [12]. It will be shown that each derivative operation of a lattice of Boolean functions creates again a lattice of Boolean functions. The created lattice can be the same lattice as the given one, but in most cases the created lattice of Boolean functions is simpler than the given lattice. Both the condition for the first case and a procedure to calculate the simpler lattice in the second case will be given. There is a direct mapping of an incompletely specified Boolean function to a lattice of Boolean functions. It will be shown that such lattices of Boolean functions are only a subclass of all lattices of Boolean functions. A unique general specification of a lattice of Boolean functions will be given.