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

TL;DR

This paper characterizes 2-threshold functions, which are conjunctions of two threshold functions, using pairs of oriented prime segments, providing a geometric perspective on their structure.

Contribution

It introduces a novel characterization of 2-threshold functions through pairs of oriented prime segments, advancing understanding of their geometric properties.

Findings

Characterization of 2-threshold functions via pairs of prime segments

Geometric interpretation of 2-threshold functions

New criteria for identifying 2-threshold functions

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.