Weighted automata are compact and actively learnable

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