Learning algebraic structures with the help of Borel equivalence relations

TL;DR

This paper explores the algorithmic learnability of algebraic structures using Borel equivalence relations, establishing a connection between learnability and reducibility to the $E_0$ relation, and proposing a descriptive set theoretic framework for analyzing learning complexity.

Contribution

It introduces a novel characterization of learnable algebraic structures via Borel reducibility to $E_0$, and suggests using descriptive set theory to analyze nonlearnable families.

Findings

Learnability characterized by reducibility to $E_0$

Framework connects algebraic learning with descriptive set theory

Analyzes the learning power of benchmark Borel equivalence relations

Abstract

We study algorithmic learning of algebraic structures. In our framework, a learner receives larger and larger pieces of an arbitrary copy of a computable structure and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if the conjectures eventually stabilize to a correct guess. We prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on reals. This motivates a novel research program, that is, using descriptive set theoretic tools to calibrate the (learning) complexity of nonlearnable families. Here, we focus on the learning power of well-known benchmark Borel equivalence relations (i.e., $E_1$, $E_2$, $E_3$, $Z_0$, and $E_{set}$).