Language Learning with a Neighbor System

We consider inductive language learning from positive examples, some of which may be incorrect. In the present paper, the error or incorrectness we consider is the one described uniformly in terms of a distance over strings. Firstly, we introduce a notion of a recursively generable distance over strings, and define a k-neighbor closure of a language L as the collection of strings each of which is at most k distant from some string in L. Then we define a k-neighbor system as the collection of original languages and their j-neighbor closures with j ≤ k, and adopt it as a hypothesis space. In ordinary learning paradigm, a target language, whose examples are fed to an inference machine, is assumed to belong to a hypothesis space without any guarantee. In this paper, we allow an inference machine to infer a neighbor closure instead of the original language as an admissible approximation. We formalize such kind of inference, and give some sufficient conditions for a hypothesis space.