  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.
