Language learnability in the limit for general metrics: a Gold-Angluin result

In his pioneering work in the field of Inductive Inference, Gold (1967) proved that a set containing all finite languages and at least one infinite language over the same fixed alphabet is not learnable in the exact sense. Within the same framework, Angluin (1980) provided a complete characterization for the learnability of language families. Mathematically, the concept of exact learning in that classical setting can be seen as the use of a particular type of metric for learning in the limit. In this short research note we use Niyogi's extended version of a theorem by Blum and Blum (1975) on the existence of locking data sets to prove a necessary condition for learnability in the limit of any family of languages in any given metric. This recovers Gold's theorem as a special case. Moreover, when the language family is further assumed to contain all finite languages, the same condition also becomes sufficient for learnability in the limit.