On teaching sets of k-threshold functions

Let $f$ be a $\{0,1\}$-valued function over an integer $d$-dimensional cube $\{0,1,\dots,n-1\}^d$, for $n \geq 2$ and $d \geq 1$. The function $f$ is called threshold if there exists a hyperplane which separates $0$-valued points from $1$-valued points. Let $C$ be a class of functions and $f \in C$. A point $x$ is essential for the function $f$ with respect to $C$ if there exists a function $g \in C$ such that $x$ is a unique point on which $f$ differs from $g$. A set of points $X$ is called teaching for the function $f$ with respect to $C$ if no function in $C \setminus \{f\}$ agrees with $f$ on $X$. It is known that any threshold function has a unique minimal teaching set, which coincides with the set of its essential points. In this paper we study teaching sets of $k$-threshold functions, i.e. functions that can be represented as a conjunction of $k$ threshold functions. We reveal a connection between essential points of $k$ threshold functions and essential points of the corresponding $k$-threshold function. We note that, in general, a $k$-threshold function is not specified by its essential points and can have more than one minimal teaching set. We show that for $d=2$ the number of minimal teaching sets for a 2-threshold function can grow as $\Omega(n^2)$. We also consider the class of polytopes with vertices in the $d$-dimensional cube. Each polytope from this class can be defined by a $k$-threshold function for some $k$. In terms of $k$-threshold functions we prove that a polytope with vertices in the $d$-dimensional cube has a unique minimal teaching set which is equal to the set of its essential points. For $d=2$ we describe structure of the minimal teaching set of a polytope and show that cardinality of this set is either $\Theta(n^2)$ or $O(n)$ and depends on the perimeter and the minimum angle of the polytope.