We show that weighted automata over the field of two elements can be exponentially more compact than non-deterministic finite state automata. To show this, we combine ideas from automata theory and communication complexity. However, weighted automata are also efficiently learnable in Angluin's minimal adequate teacher model. We include an algorithm for learning WAs over any field based on a linear algebraic generalization of the Angluin-Schapire algorithm. Together, this produces a surprising result: weighted automata over fields are structured enough that even though they can be very compact, they are still efficiently learnable.

中文翻译：

---

加权自动机紧凑并且可以主动学习

我们表明，在两个元素的域上加权自动机比非确定性有限状态自动机更紧凑。为了说明这一点，我们结合了自动机理论和通信复杂性的观点。但是，在Angluin的最小适当教师模型中也可以有效学习加权自动机。我们包括一种基于Angluin-Schapire算法的线性代数泛化来学习任意领域的WA的算法。总之，这产生了令人惊讶的结果：字段上的加权自动机足够结构化，即使它们可以非常紧凑，但仍然可以高效学习。