We study the learnability of multiplicity automata in Angluin's exact learning model , and we investigate its applications. Our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its Hankel matrix. With this theorem in hand, we present a new simple algorithm for learning multiplicity automata with improved time and query complexity, and we prove the 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 over GF(2). While multiplicity automata are shown to be useful to prove the learnability of some subclasses of DNF formulae and various other classes, we study the limitations of this method. We prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general DNF formulas or even k -term DNF for k = ω(log n ) or satisfy- s DNF formulas for s = ω(1). These results are proven by exhibiting functions in the above classes that require multiplicity automata with super-polynomial number of states.

We prove that a language of a finitely generated free monoid is regular if and only if it satisfies the positive block pumping property. This gives a positive answer to a problem posed by Ehrenfeucht, Parikh, and Rozenberg [SIAM J. Comput., 10 (1981), pp. 536--541].

The learnability of multiplicity automata has attracted a lot of attention, mainly because of its implications on the learnability of several classes of DNF formulae. The authors further study the learnability of multiplicity automata. The starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function f to the rank of a certain matrix F. With this theorem in hand they obtain the following results: a new simple algorithm for learning multiplicity automata with a better query complexity. As a result, they improve the complexity for all classes that use the algorithms of Bergadano and Varricchio (1994) and Ohnishi et al. (1994) and also obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over GF(2). They prove the learnability of some new classes that were not known to be learnable before. Most notably, the class of polynomials over finite fields, the class of bounded-degree polynomials over infinite fields, the class of XOR of terms, and a certain class of decision trees. While multiplicity automata were shown to be useful to prove the learnability of some subclasses of DNF formulae and various other classes, they study the limitations of this method. They prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general DNF formulae or even K-term DNF for k=/spl omega/ (log n) or satisfy-s DNF formulae for s=/spl omega/(1). These results are proven by exhibiting functions in the above classes that require multiplicity automata with superpolynomial number of states.

We consider the problem of identifying the behavior of an unknown automaton with multiplicity in the field Q of rational numbers (Q-automaton) from multiplicity and equivalence queries. We provide an algorithm which is polynomial in the size of the Q-automaton and in the maximum length of the given counterexamples. As a consequence, we have that Q-automata are probably approximately correctly learnable (PAC-learnable) in polynomial time when multiplicity queries are allowed. A corollary of this result is that regular languages are polynomially predictable using membership queries with respect to the representation of unambiguous nondeterministic automata. This is important since there are unambiguous automata such that the equivalent deterministic automaton has an exponentially larger number of states.