Nomura Parameters for S-Threshold Functions

Abstract

A multilevel S-threshold function characterizes a partition of a discrete point set (the domain of the function)into nonempty subsets. Each subset is labeled by one element (level) of the image set. We propose an encoding that converts a multilevel S-threshold function into a tuple of positive integers. The components of this code are generalisation of well known Chow and Nomura parameters. We derive upper bounds for the number of linear, the number of multilinear and the number of polynomial threshold functions. Comparing the encoding proposed in this paper with the one that uses discrete moments, we show that there are cases when we can get lower upper bounds for the number of considered functions.