Learning families of algebraic structures from informant

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 ${\mathbf{I}\mathbf{n}\mathbf{f}\mathbf{E}\mathbf{x}}_{\cong }$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is ${\mathbf{I}\mathbf{n}\mathbf{f}\mathbf{E}\mathbf{x}}_{\cong }$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their ${\mathrm{\Sigma }}_2^{\mathrm{i}\mathrm{n}\mathrm{f}}$-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.