Reordering the terms of a Reed-Muller or ring sum expansion of a switching function expressed in terms of the Boolean ring operations AND and EXCLUSIVE OR in a more natural way exploits the similarities between these expressions and unate functions and displays mathematical structure which apparently has not been noted before. McNaughton's n orderings on the n cube appear in a new setting and lead quickly to simple but geometrically satisfying theorems dealing with a matrix of coefficients of all 2n ring sum expressions for various polarities of inputs. The structure of these matrices for minterms, implicants, and functions are shown to have simple and attractive forms. An algorithm is presented which allows simple determination of a ring sum realization using logic array notation, and which can also be used to find minimum cost polarities. A second algorithm allows nonexhaustive and near-optimal handling of functions with DON'T CARE conditions.

Boundary points of a threshold function fare those vertices of the n-cube that produce a minimal irredundant set of inequalities for the realization of f. In this paper they are shown to be those vertices that can be contained in separating hyperplanes. With this theorem it is shown that knowledge of canonical boundary points of self-dual canonical threshold functions allows complete determination of boundary points of equivalent functions. This provides a compact and geometrically interesting characterization of threshold functions.