Inside the Muchnik Degrees I: Discontinuity, Learnability, and Constructivism

Abstract

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mind-change-bounded-learnability is equivalent to finite $(\Pi^0_1)_2$-piecewise computability (where $(\Pi^0_1)_2$ denotes the difference of two $\Pi^0_1$ sets), error-bounded-learnability is equivalent to finite $\Delta^0_2$-piecewise computability, and learnability is equivalent to countable $\Pi^0_1$-piecewise computability (equivalently, countable $\Sigma^0_2$-piecewise computability). Second, we introduce disjunction-like operations such as the coproduct based on BHK-like interpretations, and then, we see that these operations induce Galois connections between the Medvedev degree structure and associated Medvedev/Muchnik-like degree structures. Finally, we interpret these results in the context of the Weihrauch degrees and Wadge-like games.