Learning functions represented as multiplicity automata

We study the learnability of multiplicity automata in Angluin's exact learn ing model, and we investigate its applications. Our starting point is a kno wn theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its Hankel matrix. Wit h this theorem in hand, we present a new simple algorithm for learning mult iplicity automata with improved time acid query complexity, and we prove th e learnability of various concept classes. These include (among others): The class of disjoint DNF, and more generally satisfy-O(1) DNF. The class of polynomials over finite fields. The class of bounded-degree polynomials over infinite fields. The class of XOR of terms. Certain classes of boxes in high dimensions. In addition, we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials ove r GF(2). While multiplicity automata are shown to be useful to prove the learnabilit y of some subclasses of DNF formulae and various other classes, we study th e limitations of this method. We prove that this method cannot be used to r esolve the learnability of some other open problems such as the learnabilit y of general DNF formulas or even k-term DNF for k = omega(log n) or satisf y-s DNF formulas for s = omega(1). These results are proven by exhibiting f unctions in the above classes that require multiplicity automata with super -polynomial number of states.