On Exact Learning of $d$-Monotone Functions

Abstract

In this paper, we study the learnability of the Boolean class of $d$-monotone functions $f:{\cal X}\to\{0,1\}$ from membership and equivalence queries, where $({\cal X},\le)$ is a finite lattice. We show that the class of $d$-monotone functions that are represented in the form $f=F(g_1,g_2,\ldots,g_d)$, where $F$ is any Boolean function $F:\{0,1\}^d\to\{0,1\}$ and $g_1,\ldots,g_d:{\cal X}\to \{0,1\}$ are any monotone functions, is learnable in time $\sigma({\cal X})\cdot (size(f)/d+1)^{d}$ where $\sigma({\cal X})$ is the maximum sum of the number of immediate predecessors in a chain from the largest element to the smallest element in the lattice ${\cal X}$ and $size(f)=size(g_1)+\cdots+size(g_d)$, where $size(g_i)$ is the number of minimal elements in $g_i^{-1}(1)$. For the Boolean function $f:\{0,1\}^n\to\{0,1\}$, the class of $d$-monotone functions that are represented in the form $f=F(g_1,g_2,\ldots,g_d)$, where $F$ is any Boolean function and $g_1,\ldots,g_d$ are any monotone DNF, is learnable in time $O(n^2)\cdot (size(f)/d+1)^{d}$ where $size(f)=size(g_1)+\cdots+size(g_d)$. In particular, this class is learnable in polynomial time when $d$ is constant. Additionally, this class is learnable in polynomial time when $size(g_i)$ is constant for all $i$ and $d=O(\log n)$.