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.

翻译：自Angluin的开创性工作以来，通过成员查询和等价查询进行自动机的主动学习已得到广泛研究，并发展了多种推广以学习自动机的各类扩展形式。针对加权自动机，文献中已处理了若干受限情形。本文勾勒了Angluin方法（利用基于成员查询和等价查询构建的假设自动机类）在一般半环上学习加权自动机的应用边界。我们精确揭示了该方法的理论局限性，并根据函数可猜测性（对应于特定方程组解的存在性与丰富性）对其进行了分类。我们提供了存在预设形式正确假设的边界条件的语法描述。当然，从算法角度来看，知道解（大量）存在并不等同于存在有效的求解算法；最后我们讨论了一些已知条件（及其变体）以确保这一点的充分性，在若干常见半环（包括自然数）上阐释相关思想，并提出了未来研究的若干开放问题。