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

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 $Σ^0_1 \cup Π^0_1$ 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, $Σ^0_1$ can be learned adversarially, but $Π^0_1$ not); (3) some languages in $Π^0_1$ cannot be approximated by any language in $Σ^0_1$. We show corresponding results also for $Σ^0_i$ and $Π^0_i$ for arbitrary $i$.