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

02/07/2016

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by   Michael Brand, et al.

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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 ∪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.