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

Weighted automata are compact and actively learnable