Learning Equivalence Relations on Polish Spaces
Dino Rossegger, Theodore Slaman, Tomasz Steifer
TL;DR
This work develops a global framework for learning equivalence relations on Polish spaces using both explanatory and behaviorally correct paradigms. It proves that uniform and non-uniform BC- and explanatory learnability coincide and provides a precise Borel-theoretic characterization: uniform learnability corresponds to $\Sigma^0_2$ descriptions, and Borel learnability aligns with the notion of being potentially $\Sigma^0_2$. The authors establish the $\Pi^1_1$-completeness of the set of uniformly learnable equivalence relations in codes and carry out case studies showing how learnability behaves for fundamental equivalence relations arising in computability and model theory. The results connect algorithmic learning theory with forcing, higher recursion theory, and descriptive set theory, yielding both foundational insight and concrete limits for learning isomorphism-like structures. The work also highlights the landscape of learnability within the Borel hierarchy and its implications for the feasibility of complete learnable characterizations.
Abstract
We investigate natural variations of behaviourally correct learning and explanatory learning -- two learning paradigms studied in algorithmic learning theory -- that allow us to ``learn'' equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviorally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\pmbΠ^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.