Learning Union of Integer Hypercubes with Queries (Technical Report)

TL;DR

This paper introduces a polynomial-time learning algorithm for unions of integer hypercubes in fixed dimensions, with extensions for infinite hypercubes and subset queries, impacting the field of integer linear arithmetic decomposition.

Contribution

The paper presents the first polynomial-time algorithm for learning unions of integer hypercubes with queries, extending to infinite cases and practical improvements with subset queries.

Findings

Algorithm solves the problem in polynomial time for fixed dimensions.

Extensions handle infinite hypercubes and improve performance with subset queries.

Experiments show the algorithm outperforms existing methods.

Abstract

We study the problem of learning a finite union of integer (axis-aligned) hypercubes over the d-dimensional integer lattice, i.e., whose edges are parallel to the coordinate axes. This is a natural generalization of the classic problem in the computational learning theory of learning rectangles. We provide a learning algorithm with access to a minimally adequate teacher (i.e. membership and equivalence oracles) that solves this problem in polynomial-time, for any fixed dimension d. Over a non-fixed dimension, the problem subsumes the problem of learning DNF boolean formulas, a central open problem in the field. We have also provided extensions to handle infinite hypercubes in the union, as well as showing how subset queries could improve the performance of the learning algorithm in practice. Our problem has a natural application to the problem of monadic decomposition of quantifier-free integer linear arithmetic formulas, which has been actively studied in recent years. In particular, a finite union of integer hypercubes correspond to a finite disjunction of monadic predicates over integer linear arithmetic (without modulo constraints). Our experiments suggest that our learning algorithms substantially outperform the existing algorithms.