Learning Unions of Intersecting Affine Modules in One Dimension with Queries

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.