On the Use of Transeunt Triangles to Synthesize Fixed-Polarity Reed-Muller Expansions of Functions

: The transeunt triangle was originally proposed by Suprun [19] as the basis of an algorithm for synthesizing fixed-polarity Reed-Muller (FPRM) expansions of symmetric functions. However, he provided no proof that this technique produced the correct FPRM expansion. We provide such a proof, thus establishing the validity of the transeunt triangle technique. Further, we show the extent to which the transeunt triangle reduces the computational work needed. Because of the efficiency of the transeunt triangle, we are able to do experimental studies on sets of n-variable symmetric functions for large values of n never before achievable. For example, we show that a surprisingly large percentage of symmetric functions (35% for large n) are optimally realized by just two (of n+1) polarities. This is verified by exhaustive enumeration of symmetric functions with up to 31 variables and by large sample sets (1,000,000) of symmetric functions with up to 100 variables. This suggests that even greater efficiency can be achieved through a heuristic that restricts the polarities to one or both of the favored polarities.