Decomposition of group functions and the synthesis of multirail cascades

A group function is defined as a mapping from the boolean m-cube, Xm, into a finite group. When m=l, we speak of an elementary group function (or, cell). A group function f into a group H is said to be decomposable over a group G if it can be expressed as a composition of elementary group functions into G. This composition corresponds to a cascade connection of combinational cells realizing the elementary group functions, where the overall cascade realizes the group function f, in turn representing a multi-output boolean function. The basic concepts and results of Yoeli and Turner on decomposition of group functions into the Klein four-group and the alternating group of degree four are here extended to arbitrary finite groups. A useful sufficient condition for decomposability is obtained, and a general characterization derived for pairs of groups, G,H, such that all group functions into H are decomposable over G. These results are applied to the synthesis of canonical multirail logical cascade networks for the realization of r independently specified boolean functions on r rails, and comparisons are made of the efficiency of the different composition techniques that have been proposed.