# Learning families of algebraic structures from informant

## TL;DR

This paper combines computable structure theory and algorithmic learning theory to characterize when families of algebraic structures can be learned in the limit, revealing which classes are learnable based on their theories.

## Contribution

It provides a model-theoretic characterization of learnable families of structures and applies it to specific classes like lattices, Boolean algebras, and linear orders.

## Key findings

- Learnable families are characterized by their $ ext{Sigma}^{ ext{inf}}_2$-theories.
- Infinite families of distributive lattices are learnable.
- No pairs of Boolean algebras or infinite linear orders are learnable.

## Abstract

We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is $\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their $\Sigma^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable.