Learning families of algebraic structures from informant

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 ${\Sigma }_2^{\inf }$-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.

28 pages, 1 figure, forthcoming in Information and Computation