On the number of two-dimensional threshold functions

Abstract

A two-dimensional threshold function of k-valued logic can be viewed as a coloring of the points of a k × k square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics. © 2010 Society for Industrial and Applied Mathematics.