Learning Unions of Intersecting Affine Modules in One Dimension with Queries

Abstract

We study the exact learnability of finite unions of intersecting affine modules in one dimension. An affine module is a set of the form $a+\sum_{j=1}^{s}b_j \mathbb{Z}$, where $a,b_1,\ldots,b_s\in\mathbb{N}$. We say that a set definable as a finite union of affine modules is a union of intersecting affine modules if it admits a representation in which all modules have a non-empty intersection. We show that this class is efficiently exactly learnable using equivalence and subset queries. Moreover, subset queries can be replaced with membership queries when a common element is known. Our algorithm requires at most $k\log(2|x_\ell|)+2k$ counterexamples, where $k$ is the number of affine modules in the smallest representation and $x_\ell$ is the largest counterexample. This implies polynomial-time learnability in the binary representation.