The IMP game: Learnability, approximability and adversarial learning beyond $$\Sigma^0_1$$

We introduce a problem set-up we call the Iterated Matching Pennies (IMP) game and show that it is a powerful framework for the study of three problems: adversarial learnability, conventional (i.e., non-adversarial) learnability and approximability. Using it, we are able to derive the following theorems. (1) It is possible to learn by example all of Σ1 ∪ Π1 as well as some supersets; (2) in adversarial learning (which we describe as a pursuit-evasion game), the pursuer has a winning strategy (in other words, Σ1 can be learned adversarially, but Π1 not); (3) some languages in Π 0 1 cannot be approximated by any language in Σ 0 1. We show corresponding results also for Σi and Π 0 i for arbitrary i.