Learnability and Positive Equivalence Relations

TL;DR

This paper explores how positive equivalence relations influence the learnability of recursively enumerable language families, revealing that various learnability properties can be enforced or distinguished by specific relations.

Contribution

It investigates the impact of positive equivalence relations on the learnability of r.e. families, demonstrating the existence of families with diverse learnability characteristics under different relations.

Findings

Existence of r.e. families that are behaviourally correctly learnable but not vacillatorily learnable.

Existence of r.e. families that are explanatorily learnable but not confidently learnable.

A positive equivalence relation can enforce that all vacillatorily learnable families are already explanatorily learnable.

Abstract

Prior work of Gavryushkin, Khoussainov, Jain and Stephan investigated what algebraic structures can be realised in worlds given by a positive (= recursively enumerable) equivalence relation which partitions the natural numbers into infinitely many equivalence classes. The present work investigates the infinite one-one numbered recursively enumerable (r.e.) families realised by such relations and asks how the choice of the equivalence relation impacts the learnability properties of these classes when studying learnability in the limit from positive examples, also known as learning from text. For all choices of such positive equivalence relations, for each of the following entries, there are one-one numbered r.e. families which satisfy it: (a) they are behaviourally correctly learnable but not vacillatorily learnable; (b) they are explanatorily learnable but not confidently learnable; (c) they are not behaviourally correctly learnable. Furthermore, there is a positive equivalence relation which enforces that (d) every vacillatorily learnable one-one numbered family of languages closed under this equivalence relation is already explanatorily learnable and cannot be confidently learnable.