Optimal Language Learning

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 otherlearner that: requires as littleof each presentation of each object in the class in order to identify that object, and, for somepresentation of someobject in the class, requires lessof that presentation in order to identify that object. Wiehagen considered this notion in the context of functionlearning, and characterizedan 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 languagelearning. Intuitively, a language learner identifies a class of languages optimally iff there is no other learner that: requires as little of each textfor 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 nothold in this setting. Specifically, optimality is notsufficient to guarantee Wiehagen's conditions; though, those conditions aresufficient to guarantee optimality. Second, it is shown that the failure of this analog is notdue 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 notrestrict 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 $\mathcal {K}$.