On the learning power of Friedman-Stanley jumps

TL;DR

This paper explores the learning capabilities of Friedman-Stanley jumps in the context of algebraic structures and descriptive set theory, revealing their ability to distinguish countable structures via infinitary sentences.

Contribution

It characterizes the learning power of finite Friedman-Stanley jumps and introduces new methods for analyzing the complexity of Borel equivalence relations.

Findings

Friedman-Stanley jumps can learn families of countable structures distinguished by infinitary sentences.

The paper establishes the continuous reducibility of certain equivalence relations to these jumps.

New techniques are developed for assessing the complexity of Borel equivalence relations.

Abstract

Recently, a surprising connection between algorithmic learning of algebraic structures and descriptive set theory has emerged. Following this line of research, we define the learning power of an equivalence relation $E$ on a topological space as the class of isomorphism relations with countably many equivalence classes that are continuously reducible to $E$. In this paper, we describe the learning power of the finite Friedman-Stanley jumps of $=_{\mathbb{N}}$ and $=_{\mathbb{N}^\mathbb{N}}$, proving that these equivalence relations learn the families of countable structures that are pairwise distinguished by suitable infinitary sentences. Our proof techniques introduce new ideas for assessing the continuous complexity of Borel equivalence relations.