Optimal Language Learning

Abstract

Gold’s original paper on inductive inference introduced a notion of an optimal learner. Intuitively, a learner identifies a class of objects optimally iff there is no other learner that: requires as little of each presentation of each object in the class in order to identify that object, and, for some presentation of some object in the class, requires less of that presentation in order to identify that object. Wiehagen considered this notion in the context of function learning, and characterized an optimal function learner as one that is class-preserving, consistent, and (in a very strong sense) non-U-shaped, with respect to the class of functions learned.

Herein, Gold’s notion is considered in the context of language learning. Intuitively, a language learner identifies a class of languages optimally iff there is no other learner that: requires as little of each text for each language in the class in order to identify that language, and, for some text for some language in the class, requires less of that text in order to identify that language.

Many interesting results concerning optimal language learners are presented. First, it is shown that a characterization analogous to Wiehagen’s does not hold in this setting. Specifically, optimality is not sufficient to guarantee Wiehagen’s conditions; though, those conditions are sufficient to guarantee optimality. Second, it is shown that the failure of this analog is not due to a restriction on algorithmic learning power imposed by non-U-shapedness (in the strong form employed by Wiehagen). That is, non-U-shapedness, even in this strong form, does not restrict algorithmic learning power. Finally, for an arbitrary optimal learner F of a class of languages $\mathcal {L}$, it is shown that F optimally identifies a subclass $\mathcal {K}$ of $\mathcal {L}$ iff F is class-preserving with respect to $\mathcal {K}$.