Learning Theory in the Arithmetic Hierarchy

TL;DR

This paper analyzes the computational complexity of index sets for various learnability criteria within the arithmetic hierarchy, establishing exact classifications and revealing new insights into the limits of learnability.

Contribution

It determines the precise arithmetic complexity of index sets for multiple learning criteria and proves a novel $ ext{Sigma}_5^0$-completeness result for behaviorally correct learning.

Findings

Exact complexity classifications for finite, limit, behaviorally correct, and anomalous learning.

Proof of $ ext{Sigma}_5^0$-completeness for behaviorally correct learning.

Existence of a $ ext{Delta}_2^0$ enumeration witnessing failure for non-learnable families.

Abstract

We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning, learning in the limit, behaviorally correct learning and anomalous learning in the limit. In proving the $\Sigma_5^0$-completeness result for behaviorally correct learning we prove a result of independent interest; if a uniformly computably enumerable family is not learnable, then for any computable learner there is a $\Delta_2^0$ enumeration witnessing failure.