On the properties of multiple-valued functions that are symmetric in both variable values and labels

Functions that are symmetric in both variable labels and variable values are useful as benchmarks for logic minimization algorithms. We present the properties of such functions, showing that they are isomorphic to partitions on n (the number of variables) with no part greater than r (the number of logic values). From this, we enumerate these functions. Further we derive lower bounds, upper bounds, and exact values for the number of prime implicants in the minimal sum-of-products expressions for certain subclasses of these functions.