Learning Behaviors of Automata from Multiplicity and Equivalence Queries

We consider the problem of identifying the behavior of an unknown automaton with multiplicity in the field $\Ratviii$ of rational numbers ($\Ratviii$-automaton) from multiplicity and equivalence queries. We provide an algorithm which is polynomial in the size of the $\Ratviii$-automaton and in the maximum length of the given counterexamples. As a consequence, we have that $\Ratviii$-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.