A Faster Algorithm for Learning Decision Trees With Module Nodes

One of the challenging problems regarding learning decision trees is whether decision trees over the domain Z~, in which each internal node is labeled by a module function axi(mod N) for some i E (1 .... ,n), are efficiently learnable with equivalence and membership queries [5]. Given any decision tree T, let s be the number of its leaf nodes. Let N = p~l...p~ be its prime decomposition, and let 7(N) = (tl 1)... (t~ + 1). We show that when all the module functions in T have the same coefficient, then there is a polynomial time algorithm for learning T using at most s(s + 1)7(N) equivalence queries and at most s2nT(N) membership queries. Our algorithm is substantially more efficient than the algorithm designed in [7] for learning such decision trees. When N is a prime, our algorithm implies the best-known algorithm in [4] for learning decision trees over a finite alphabet.