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

TL;DR

This paper generalizes Gold's classical result on language learnability in the limit by establishing a necessary condition for learnability under any metric, and shows it is also sufficient when all finite languages are included.

Contribution

It introduces a unified metric-based framework for language learnability, extending Gold's theorem and providing new necessary and sufficient conditions.

Findings

Generalizes Gold's theorem to arbitrary metrics

Establishes a necessary condition for learnability in the limit

Shows sufficiency when the language family includes all finite languages

Abstract

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.