Derivative Operations for Lattices of Boolean Functions

This paper explores derivative operations of the Boolean differential calculus for lattices of Boolean functions. Such operations are needed to design circuits with short delay and low power consumption [3] as well as to calculate minimal complete sets of fitting test patterns [4]. 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. There is a direct mapping of an incompletely specified Boolean function to a lattice of Boolean functions. We will show 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.