Learning equivalence relations on Polish spaces

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 behaviourally correct and explanatory learnable equivalence relations coincide both in uniform and nonuniform 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 ∏¹₁-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.