Some New Types of Logical Completeness

In this paper, we consider the problem of determining if a given set of simple combinational functions, called logic primitives, can be used to realize any arbitrarily complex combinational function. The complex combinational function is realized by suitably interconnecting the primitives, and in most cases several copies of the primitives. Such a set of primitives is said to be complete in some sense. Two important types of logical completeness that have been previously studied are: 1) strong completeness where the circuit inputs are the uncomplemented variables x1,x2,···,xn, and 2) weak completeness where the circuit inputs are 0,1,x1,x2,···,xn. In this paper, we discuss two new types of logical completeness: strong c-completeness with inputs x1, x̄1, x2, x̄2,···,xn, x̄n and weak c-completeness with inputs 1, 0, x1, x̄1, x2, x̄2,···, xn, x̄n, and we derive necessary and sufficient conditions for a set of primitives to be strong c-complete or weak c-complete. We also discuss the relations between the above four types of completeness. Finally, we consider two more types, complement completeness and dual completeness, which may have applications to some recent technologies.