8th Annual Symposium on Switching and Automata Theory (SWAT 1967) (1967)
Oct. 18, 1967 to Oct. 20, 1967
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.1967.10
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.
H. S. Stone and B. Elspas, "Decomposition of group functions and the synthesis of multirail cascades," 8th Annual Symposium on Switching and Automata Theory (SWAT 1967)(FOCS), Texas, 1967, pp. 184-196.