A Spectral Algorithm for Ternary Function Classification

The spectral representation and classification of 2-valued and multiple-valued functions has been previously studied and found to be useful in logic design and testing for conventional circuits. Spectral techniques also have potential application for reversible and quantum circuits. This paper addresses the classification of ternary functions into spectral translation equivalence classes. An efficient algorithm is presented that determines the spectral translations to map a given function to the representative function for the equivalence class containing the given function. Using this algorithm we show that the 2-variable ternary functions partition into 11 equivalence classes. While the number of spectral equivalence classes for ternary functions with 3 or more variables is very large, prohibiting full enumeration, we determine a lower bound of 167,275 classes for 3 variables. The algorithm can be used for a significant number of variables to quickly determine if two functions fall within the same equivalence class and, if they do, to find a sequence of spectral translations to map one to the other. Generalization of the approach to higher radix functions is briefly discussed.