In this paper, we present a new algorithm that can identify in polynomial time and data using positive examples any class of subsequential functions that share a particular finite-state structure. While this structure is given to the learner a priori, it allows for the exact learning of partial functions, and both the time and data complexity of the algorithm are linear. We demonstrate the algorithm on examples from natural language phonology and morphology in which the needed structure has been argued to be plausibly known in advance. A procedure for making any subsequential transducer onward without changing its structure is also presented.