Characterization of Generalized S-Threshold Functions by Nomura Parameters.

This paper deals with the encoding and enumerating of generalized S-threshold functions. Such a function performs a partition of its finite domain set, where induced subsets are separable with parallel hyper-surfaces. Each hyper-surface is characterized as a linear combination of functions from the fixed tuple S. We show that if the function is a generalized S-threshold then it is uniquely characterized by the encoding presented here. The parameters of the code can be seen as a direct generalization of well known Chow and Nomura parameters. Using this characterization, we obtain upper bounds for the number of functions from three classes of threshold functions: linear, multilinear and polynomial with a restricted degree. We give the correspondence of our encoding with one existing in the literature that uses discrete moments. It is shown that, in some cases, we get a sharper upper bounds for the number of threshold functions than those derived using discrete moments.