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.

翻译：Gold(1967年)在引导推论领域的开拓性工作中证明,一套包含所有有限语言和至少一种固定字母上的一种无限语言的套套套在确切意义上是无法学懂的,在同一框架内,Angluin(1980年)为语言家庭的学习提供了完整的特征描述,从数学角度讲,古典环境中的精确学习概念可被视为在限度内使用某种特定类型的教学指标。在这个简短的研究说明中,我们使用Niyogi的布卢姆和布卢姆(1975年)关于存在锁定数据集以证明在任何特定标准中任何语言家庭范围内学习的必要条件的扩展版理论。这恢复了Gold的理论,此外,如果进一步假定语言家庭包含所有有限的语言,同样条件也足以在限度内学习。