On the learning power of Friedman-Stanley jumps

Recently, a surprising connection between algorithmic learning of algebraic structures and descriptive set theory has emerged. Following this line of research, we define the learning power of an equivalence relation $E$ on a topological space as the class of isomorphism relations with countably many equivalence classes that are continuously reducible to $E$. In this paper, we describe the learning power of the finite Friedman-Stanley jumps of $=_{\mathbb{N}}$ and $=_{\mathbb{N}^\mathbb{N}}$, proving that these equivalence relations learn the families of countable structures that are pairwise distinguished by suitable infinitary sentences. Our proof techniques introduce new ideas for assessing the continuous complexity of Borel equivalence relations.