Limit Learning Equivalence Structures

TL;DR

This paper characterizes which families of computable equivalence structures can be learned from informant data, establishing complexity bounds and exploring variants like learning from text and finite learning.

Contribution

It provides a complete characterization of InfEx-learnability for equivalence structures and links structure learning to language learning tasks.

Findings

Characterization of InfEx-learnable families of structures

Bound of  on computational complexity for learning families

Analysis of learning variants including text and finite learning

Abstract

While most research in Gold-style learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pairs of elements of a structure are related and which are not. The aim of the learner is to find (an effective description of) the isomorphism type of the structure presented in the limit. In accordance with language learning we call this learning criterion InfEx-learning (explanatory learning from informant). Our main contribution is a complete characterization of which families of equivalence structures are InfEx-learnable. This characterization allows us to derive a bound of $\mathbf{0''}$ on the computational complexity required to learn uniformly enumerable families of equivalence structures. We also investigate variants of InfEx-learning, including learning from text (where the only information provided is which elements are related, and not which elements are not related) and finite learning (where the first actual conjecture of the learner has to be correct). Finally, we show how learning families of structures relates to learning classes of languages by mapping learning tasks for structures to equivalent learning tasks for languages.