On the learning power of Friedman-Stanley jumps

TL;DR

This work develops a learning-theoretic framework for classifying countable structures via continuous reductions, focusing on the finite Friedman–Stanley jumps of $=_{\mathbb{N}}$ and $=_{\mathbb{N}^{\mathbb{N}}}$. It establishes a precise syntactic characterization: a countable family of pairwise nonisomorphic structures is $=^{n+}_{\mathbb{N}}$-learnable (resp. $=^{n+}_{\mathbb{N}^{\mathbb{N}}}$-learnable) iff the structures are distinguishable by $\Sigma_{2n-1}^{\mathrm{inf}}$-sentences (resp. $\Sigma_{2n}^{\mathrm{inf}}$-sentences). Central to the results are the new tools: nice$_{n,\varphi}$ functions that force a structure to present a canonical full or initial witness, and permission operators that preserve these witnessing patterns across higher levels. A compactness result shows that pairwise learnability suffices to guarantee learnability of any countable family, enabling a robust transfer from finite to infinite families. The findings illuminate the continuous complexity of Borel equivalence relations, yield a chain of learning powers of length $\omega$, and suggest a scalable method to analyze learning power for broader classes of 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.

Theorems & Definitions (35)

• Theorem 1.1
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Lemma 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9