A characterization of 2-threshold functions via pairs of prime segments

Abstract

A $\{0,1\}$-valued function on a two-dimensional rectangular grid is called threshold if its sets of zeros and ones are separable by a straight line. In this paper we study 2-threshold functions, i.e. functions representable as the conjunction of two threshold functions. We provide a characterization of 2-threshold functions by pairs of oriented prime segments, where each such segment is defined by an ordered pair of adjacent integer points.