Feasability of Learning Weighted Automata on a Semiring

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.