On the minimal teaching sets of two-dimensional threshold functions

TL;DR

This paper derives exact formulas for counting minimal teaching sets of 2D threshold functions, refines these counts for size 3 sets, and analyzes their average size, with implications for line arrangements in the plane.

Contribution

It provides exact enumeration formulas for minimal teaching sets of 2D threshold functions and analyzes their average size, advancing understanding of their combinatorial structure.

Findings

Exact formulas for the number of threshold functions with minimal teaching sets of size 3 or 4.

Refined counts specifically for minimal teaching sets of size 3.

Asymptotic average size of minimal teaching sets is 3.5.

Abstract

It is known that a minimal teaching set of any threshold function on the twodimensional rectangular grid consists of 3 or 4 points. We derive exact formulae for the numbers of functions corresponding to these values and further refine them in the case of a minimal teaching set of size 3. We also prove that the average cardinality of the minimal teaching sets of threshold functions is asymptotically 7/2. We further present corollaries of these results concerning some special arrangements of lines in the plane.