The Characterization and Properties of Cascade Realizable Switching Functions

This paper is a study of switching functions realizable by a single cascaded switching network composed of two-input, one-output elements where a distinct variable is applied to one input and the other receives the output of the previous element. A special type of Boolean formula called a standard cascade form is introduced with the property that all cascade realizable functions, and only these, can be written in this form. This characterization leads to a strong necessary condition on such functions: there is no consensus for any pair of its prime implicants which are, therefore, all core. It also leads to a new, efficient procedure for testing an arbitrary function f for cascade realizability. The test operates on the prime implicants of f, and yields a realization employing a particular complete set of cell types from which it is especially easy to derive any other realization using any other complete set.