  Since the seminal work by Angluin, active learning of automata, by membership
and equivalence queries, has been extensively studied and several
generalisations have been developed to learn various extensions of automata.
For weighted automata, restricted cases have been tackled in the literature and
in this paper we chart the boundaries of the Angluin approach (using a class of
hypothesis automata constructed from membership and equivalence queries)
applied to learning weighted automata over a general semiring. We show
precisely the theoretical limitations of this approach and classify functions
with respect to how guessable they are (corresponding to the existence and
abundance of solutions of certain systems of equations). We provide a syntactic
description of the boundary condition for a correct hypothesis of the
prescribed form to exist. Of course, from an algorithmic standpoint, knowing
that (many) solutions exist need not translate into an effective algorithm to
find one; we conclude with a discussion of some known conditions (and variants
thereof) that suffice to ensure this, illustrating the ideas over several
familiar semirings (including the natural numbers) and pose some open questions
for future research.
