Uniformity in learning structures
Vittorio Cipriani, Dino Rossegger
TL;DR
This paper investigates what happens to Ex-learning and BC-learning for countable structures when the usual requirement of pairwise nonisomorphism is dropped. It proves that Ex-learning and BC-learning remain equivalent in this broader setting, though the translation from BC- to Ex-learners is non-uniform, becoming uniform only when the learning family is provided as input. The work connects learnability to back-and-forth relations, offers a syntactic characterization via $\Sigma^{\mathrm{in}}_{2}$ quasi-Scott sentences, and analyzes the descriptive-set-theoretic complexity of learnable families, showing $\mathbf{Rep}$ is $\Sigma^1_1$-complete while $\mathbf{NoRep}$ is Borel and no global Borel reduction exists between them. It also surveys the role of trees, analytic sets, and pseudo well-orders in encoding and understanding the limits of learnability and uniformity in this context.
Abstract
The standard framework for studying learning problems on algebraic structures assumes that the structures in the target family are pairwise nonisomorphic. Under this assumption, the most widely investigated learning criterion--Ex-learning--becomes inherently equivalent to the well-known paradigm of Bc-learning. This paper explores what happens when the nonisomorphism requirement is removed and analyzes the extent to which these two learning criteria remain uniformly equivalent.