On the number of two-dimensional threshold functions

A two-dimensional threshold function of k-valued logic can be viewed as coloring of the points of a k x 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.Comment: 17 pages, 2 figure

Computational Geometry of Linear Threshold Functions

Linear threshold machines are defined to be those whose computations are based on the outputs of a set of linear threshold decision elements. The number of such elements is called the rank of the machine. An analysis of the computational geometry of finite-rank linear threshold machines, analogous to the analysis of finite-order perceptrons given by Minsky and Papert, reveals that the use of such machines as "general purpose pattern recognition systems" is severely limited. For example, these machines cannot recognize any topological invariant, nor can they recognize non-trivial figures "in context"