A high-speed parallel adder of digitally represented numbers called the conditional-sum early completion adder (CSCA) will be described. The CSCA design is based on the computation of "conditional" sums, carries, and column completion detection logic.

In previous papers, Post, Webb, Götlind and the present author have described some Sheffer functions (in Swift's terminology, “independent binary generators”) in m-valued logic. Professor J. Dean Swift has recently isolated the symmetric Sheffer functions of 3-valued logic. In the present paper, we will prove some properties of Sheffer functions in m-valued logic and isolate all of the Sheffer functions of 3-valued logic. Before we proceed we will define some terms which we will find convenient. A set of functions in m-valued logic is functionally complete, if the set of the functions which can be defined explicitly from the functions of the set is exactly the set of all functions of m-valued logic. A function is functionally complete, if its unit set is functionally complete. A Sheffer function is a two-place functionally complete function. If i and j are truth values (1 i, j ≤ m), we will say i ~ j (D), if D is a decomposition of the truth values 1, …, m into 2 or more disjoint non-empty classes and i and j are elements of the same class. A binary function f(p, q) satisfies the substitution law for a decomposition D, if for any truth values h, i, j, k, whenever h ~ j (D) and i~k(D), then f(h, i) ~ f(j, k) (D). The function f(p,q) satisfies the co-substitution law for D, if for any truth values h, i, j, k, whenever f(h, i) ~ f(j, k) (D), then h ~ j (D) or i ~ k (D). We will say f(p, q) has the proper substitution property, if there is a decomposition of the truth values into less than m classes for which it satisfies the substitution law.