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 in a number of queries that is polynomial in the size of the minimal weighted automaton. 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.

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，在许多以最小加权自动机大小为多项式的查询中，也可以在Angluin的最小适当教师模型中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何领域的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，即使它们非常紧凑，它们仍然可以高效学习。