Remarks on Applicability of Spectral Representations on Finite Non-Abelian Groups in the Design for Regularity

In several publications, the use of non-Abelian groups has been suggested as a method to derive compact representations of logic functions. The compactness has been measured in the number of product terms in the case of functional expressions and the number of nodes, the width, and the interconnections in the case of decision diagrams. In this paper, we discuss Fourier representations on finite non-Abelian groups in synthesis for regularity. The initial domain group for a logic function (binary or multiple-valued) is replaced by a non-Abelian group by encoding of variables. The function is then decomposed into matrix-valued Fourier coefficients, that are easy to implement as building blocks over a technological platform with regular structure. We point out that spectral representation of non-Abelian groups is capable of capturing regularities in functions and transferring them in the spectral domain. In many cases, weak regularities in the original domain are converted into much stronger regularities in the spectral domain due to the regular structure of unitary irreducible group representations upon which the Fourier expressions are based.