# Isomorphism problems and learning of algebraic structures

## Vittorio Cipriani

In this talk, we study the classification of isomorphism problems with countably many isomorphism types. Borel reducibility has proven to be a powerful tool for classifying classes of countable structures under isomorphism; however, such problems, when restricted to classes with only countably many isomorphism types, have the same Borel complexity.

To nicely classify such problems, we present a framework defined in a series of papers by Bazhenov, Fokina, Kötzing, and San Mauro which combines ideas from computable structure theory and algorithmic learning theory. Here, given a countable family of structures \(\mathfrak{K}\) a **learner** receives larger and larger pieces of an isomorphic copy of a structure in \(\mathfrak{K}\) 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.

Together with Bazhenov and San Mauro, we showed that a countable family of countable structures is learnable if and only if the corresponding isomorphism problem reduces to \(E_0\), the equivalence relation of eventual agreement on infinite binary sequences. Replacing \(E_0\) with other Borel equivalence relations, one obtains a hierarchy to rank such isomorphism problems. This improves the aforementioned framework by providing a way of calibrating the complexity of non-learnable families.

Starting from a result of Bazhenov, Fokina, and San Mauro we give a model-theoretic characterization of the learning power of several equivalence relations (with a focus on Friedman-Stanley jumps of the identity on \(2^\mathbb{N}\) and \(\mathbb{N}\)) and also of other classical learning criteria.

This talk collects joint works with Bazhenov, Jain, Marcone, San Mauro, and Stephan.