On characterizations of learnability with computable learners

We study computable PAC (CPAC) learning as introduced by Agarwal et al. (2020). First, we consider the main open question of finding characterizations of proper and improper CPAC learning. We give a characterization of a closely related notion of strong CPAC learning, and provide a negative answer to the COLT open problem posed by Agarwal et al. (2021) whether all decidably representable VC classes are improperly CPAC learnable. Second, we consider undecidability of (computable) PAC learnability. We give a simple general argument to exhibit such undecidability, and initiate a study of the arithmetical complexity of learnability. We briefly discuss the relation to the undecidability result of Ben-David et al. (2019), that motivated the work of Agarwal et al.